These are the instructor’s lecture notes for a course in Trigonometry
covering rectangular vs polar coordinate systems,
some synthetic trigonometry,
some analytic geometry regarding the graphs of trigonometric functions,
and some vector geometry in two- and three-dimensional space.
Start each lecture with a "motivating question".
End each lecture with a “Before next class, review … ”
Is this lesson meaningfully enhanced with a Desmos geometry demo?
Is this lesson meaningfully enhanced with an historical aside?
Conventions & Preliminaries
Defining a function \(f,\)
the notation \(f(x)\) is read as “\(f\) of \(x\)”
and represents the output that results
from evaluating \(f\) at the input \(x,\)
generally expressed as a formula.
The functions most often used in this course
will be those conventionally called trigonometric
and inverse trigonometric functions.
The notation \(\mathtt{trig}\) will be used to represent
a generic trigonometric function
— any of \(\sin\) or \(\cos\) or \(\tan\)
or \(\csc\) or \(\sec\) or \(\cot\) —
and instead of \(\mathtt{trig}^{-1}\) which is typical in the US
we’ll use \(\mathrm{arc}\mathtt{trig}\) to represent
a generic “inverse” trigonometric function.
Sometimes a symbol will be overloaded to represent
both a thing and the measure of that thing.
For example the symbol \(A\) may be used to represent
a side of a triangle and the measure (length) of that side.
Another example, the symbol \(\theta\) may be used to represent an angle
and the measure of that angle.
This is a subtly that folks not trained in mathematics likely don’t consciously consider,
and the risk of any subconscious dissonance this creates in the mind of students is outweighed
by the benefits of tidier notation;
it would be cumbersome, and make for ugly formulas, to write
\(\operatorname{m}\bigl(\theta\bigr)\) for the measure of \(\theta.\)
Characters typeset in a roman font like “\(\mathrm{e}\)” or “\(\mathbf{j}\)”
or “\(\operatorname{proj}\)” represent specific constants or named functions,
whereas italicized characters like “\(x\)” or “\(f\)” are indeterminate variables or functions.
Bold characters like \(\bm{v}\) are vector-valued,
whereas non-bold characters like \(v\) will be scalar-valued (numeric).
Variables representing angles will be denoted by lowercase Greek letters,
usually \(\theta\) and \(\varphi,\)
or sometimes \(\alpha\) and \(\beta\) and \(\gamma.\)
Rectangular Geometry Fundamentals
How do we describe the location of a point in the plane?
How does geometry work in two-dimensional space?
Describe “the plane”.
Draw the origin and coordinate axes.
This imbues the plane with a
rectangular (Cartesian) coordinate system
and we now call it the \(xy\)-plane.
Think of the plane as having a “grid” geometry.
Label the quadrants.
Plot the points \((4,8)\) and \((2,-3)\).On the whiteboard, draw some coordinate axes, poke a random point, and use a meter stick to determine its coordinates.
The “shape” of a space is determined in part
by how distance is defined; this is a choice we’re making.
Our choice of distance between points \(A\) and \(B\)
with coordinates \(\bigl(x_1, y_1\bigr)\) and \(\bigl(x_2, y_2\bigr)\) respectively
will be \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}.\)
This based on the Pythagorean theorem.
Pythagoras was a cult leader btw.
Calculate distance between \((4,8)\) and \((2,-3).\)
The midpoint \((\overline{x}, \overline{y})\) between two points
\(\bigl(x_1, y_1\bigr)\) and \(\bigl(x_2, y_2\bigr)\)
is exactly what it sounds like, and those coordinates are given by the formulas
\(\overline{x} = \frac{1}{2}(x_1+x_2)\) and \(\overline{y} = \frac{1}{2}(y_1+y_2);\)
it’s just the averages of the coordinates of the endpoints.
Calculate midpoint of the segment between \((4,8)\) and \((2,-3).\)
The slope of the line containing those two points is
\( m = \frac{\text{rise}}{\text{run}} = \frac{y_2-y_1}{x_2-x_1}\,. \)
This is the “rate” at which the line is increasing/decreasing.
It's also referred to as the pitch or grade.
The “point-slope” form, “slope-intercept” form,
and “general” form of a line are respectively are
\[ (y-y_1) = m(x-x_1) \qquad y = mx+b \qquad Ax + By = C \,, \]
but these names are stupid.
What’s an equation for the line with \(y\)-intercept 5 and slope \(\frac{2}{3}?\) What’s an equation for the line with slope \(\frac{2}{3}\) passing through the point \((4,8)?\) What’s an equation for the line passing through the points \((4,8)\) and \((2,-3)?\)
\(y = \frac{11}{2}x-14 \)
Horizontal and vertical lines
have special forms, \(y=b\) and \(x=a\) respectively.
Two lines are parallel
if they never intersect, and so have the same slope.
Two lines are perpendicular
(or orthogonal, or normal to one another)
if they intersect at a right angle,
so one’s slope will be the negative reciprocal of the other.
What’s an equation for the line parallel to \(y = -5x + 3\) that passes through the point \((2,-3)?\) What’s an equation for the line perpendicular to \(y = -5x + 3\) passing through \((2,-3)?\) What’s the distance from the point \((2,-3)\) to the line \(y = -5x + 3\)
A circle is a curve defined as all points
that are a fixed distance, its radius,
from a given point, its center.
What’s the distance between \((4,8)\) and \((x,y)?\)
What’s an equation for all points \((x,y)\)
such that the distance between \((4,8)\) and \((x,y)\) is 7?
The equation corresponding to a circle
is the distance formula with one of the points kept arbitrary.
I.e. \((x-h)^2 + (y-k)^2 = r^2\) is a circle
with radius \(r\) and center \((h,k).\)
A circle with radius \(r\)
has circumference (perimeter) \(2 \pi r\)
and area \(\pi r^2,\) where
\({\pi \approx 3.1415926535897932384626433832795028841968.}\)
The unit circle is the specific circle
with radius 1 that’s centered at the origin.
What’s an equation for the unit circle? Where does the unit circle intersect the line \(y = \sqrt{3}x?\)
Angle Measure + Technology Demo
Bring a straightedge and giant protractor for demonstration.
Draw and define angles, initial side and terminal side.
Positive angles are clockwise — right-hand rule.
Degree measure: most commonly used, and what we’ll use.
Based on Babylonian base-60 “sexigesimal” math and ~365 days in a year.
Cut a full rotation into 360 “pieces” each called a degree (1°).
Always put the degree sign.
A full rotation is \(360°,\) a half a rotation is \(180°,\)
a quarter rotation, a right angle is \(90°.\)
A eighth of a rotation, a “diagonal” angle is \(45°.\)
It’s helpful to think in terms of fractions of a rotation.
Draw an angle on the whiteboard
and literally measure it with the giant protractor.
Fractional degrees: either a decimal number of degrees (DD)
or degrees/minutes/seconds (DMS),
which is more often used in terrestrial navigation, GPS.
How do you express 123.456789° in degrees/minutes/seconds?
How many degrees is 33°22′11″ expressed as a decimal?
Radian measure: more common among scientists,
and lends itself nicely to geometry.
Imagine an angle corresponding to a sector
of a circle of radius \(r.\)
We say the arc subtends and angle (or sector),
and the angle (or sector) is subtended by the arc.
The radian measure of an angle is
the number of radii that fit along that arc.
I.e. for an angle of radian measure \(\theta\)
the length of the arc subtending the angle will be \(r\theta\)
and the area of the sector will be \(\frac{1}{2}r^2\theta.\)
A full rotation is \(2\pi\) radians; the full circumference.
A half rotation is \(\pi\) radians.
A quarter rotation, a right angle, \(\frac{\pi}{2}\) radians.
It’s helpful to think in terms of fractions of a rotation.
How many degrees is \(\pi/4\) radians?
How many degrees is \(\pi/3\) radians?
How many degrees is \(\pi/6\) radians?
How many degrees is 1 radian?
How many radians is 77°?
Sometimes an angle is negative, or encompasses a full rotation or more,
but we want to talk about the angle within one rotation it corresponds to.
Define coterminal angles and principal angles.
What is the degree measure
of the principal angle coterminal to 1776°?
Note 1776° is coterminal to -24°.
Finally a few geometric considerations to be aware of.
Similarity: scaling changes lengths but preserves angles.
Two angles \(\theta\) and \(\varphi\) are complementary
if \(\theta + \varphi = 90°;\)
the two non-right angles in a right triangle are complements.
Two angles \(\theta\) and \(\varphi\) are supplementary
if \(\theta + \varphi = 180°,\) if they together lie along a straight line.
Given two parallel lines and a traversal line that passes through them,
given the measurement of one angle
we can determine the measures of all the others.
Technology: demonstrate how to,
Convert between decimal degrees (DD)
and degrees/minutes/seconds (DMS)
on a TI-30XIIS and on a TI-83/84+,
how to use these calculators better (storing variables),
how to switch between radian mode and degree mode,
use both geometry Desmos and graphing calculator Desmos.
Review rectangular coordinates before next lecture.
Polar Geometry Fundamentals
Bring giant protractor and class protractors
for demonstration and exercises.
Is there a better way to describe the location of a point in the plane?
Rectangular coordinate review before introducing polar coordinates.
Define pole and polar axis and normal axis
This imbues the plane with a polar coordinate system.
Right-hand rule again.
Think of the plane as having a “spider web” or “radar” grid.
Plot the points \((6,27°)\) and \((2,-55°)\).
Draw coordinate axes and a point on the whiteboard
and use the giant protractor
to literally measure the polar coordinates of that point.
Carefully demonstrate how to convert between
rectangular and polar coordinates.
Think of sine and cosine as percentages.
Arctangent, an “inverse” of tangent,
takes a slope and returns and angle.
It’s the tan-1 button on your calculator.
The word has come with some distortion
from Sanskrit through Arabic and Latin.
Accounts differ on the details but the basic story is this:
the Sanskrit jya ("chord") was taken into Arabic as jiba
but the word that was translated into Latin was not this word but jaib ("bay")
and this became sinus ("bay" or "curve") which was anglicized as sine.
What are the polar coordinates of the point
with rectangular coordinates \((1,2)?\)
What are the rectangular coordinates of the point
with polar coordinates \((6,27°)?\)
What are the polar coordinates of the point
with rectangular coordinates \((-7,3)?\)
Mention atan2(y,x).
For a given point there are infinitely many pairs
of polar coordinates that describe that point.
E.g. (1,23°) and (1,383°) and (1,-337°) and (-1,203°)
all describe the same point,
but (1,23°) is “principal” among them since it’s simplest in a sense.
For a given point only one pair of polar coordinates \((r,\theta)\)
with \(r \geq 0\) and \(0 \leq \theta \lt 360°\) that describes it.
What are the “principal” polar coordinates of the point
with point coordinates \((3,1776°)?\)
Typographic note:
some authors write their trig functions without parenthesis;
know that \(\sin \theta\) and \(\sin\bigl(\theta\bigr)\) are the same.
Also it’s conventional to write \(\sin^2\bigl(\theta\bigr)\)
for \(\bigl(\sin\theta\bigr)^2.\)
However \(\tan^{-1}\bigl(\theta\bigr) \neq \bigl(\tan(\theta)\bigr)^{-1};\)
a power of \(-1\) on a function denotes its inverse function,
whereas the latter expression represents a reciprocal.
When \(r = 1,\)
because they’re the \(x\)- and \(y\)-coordinate on a circle of radius one,
\(\sin^2\bigl(\theta\bigr) + \cos^2\bigl(\theta\bigr) = 1.\)
This is called a Pythagorean identity.
“Unit Circle” Trigonometry
Give pop quiz on day two.
The unit circle is the circle of radius one
centered at the origin.
Its circumference is \(2\pi.\)
Show that the point
\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)
is on the unit circle.
Helpful to find the sine and cosine of \(30°\) later.
Per the discussion of converting between polar and rectangular coordinates,
whereas on a circle of radius \(r\)
the sine and cosine functions give you percentages,
the factors by which you scale \(r\) to get the \(y\)- and \(x\)-coordinate
on that circle corresponding to a given angle,
on the unit circle we can think of the values of sine and cosine
as the literal coordinates themselves.
This can be helpful.
What are the coordinates of the point on the unit circle
corresponding to angles 90° or -180° or 900°?
Start a running “unit circle” up on the white board.
Sometimes finding the coordinates of this point,
i.e. finding the values of sine and cosines, is easy,
but there are other angles for which this question
requires some clever geometry to answer.
What are the coordinates of the point on the unit circle
corresponding to the angles
\(45°?\) or \(30°?\) or \(60°?\)
Note that these are just nice fractions of a rotation.
Add them to the unit circle on the board.
Gotta refer to simply symmetry for \(45°.\)
Note that \(1/\sqrt{2} = \sqrt{2}/2.\)
The coordinate for \(30°\) are actually a bit tricky:
note that the point \((x,y)\) for \(30°\)
has the property that it’s equidistant to
\((0,1)\) and to \((x,-y),\)
which means \(2y\) equals \(\sqrt{(x-0)^2+(y-1)^2}.\)
Solve this equation simultaneously with \(x^2+y^2=1.\)
Then argue \(60°\) from symmetry.
It is worthwhile to memorize
these “nice” input-output pairs for sine and cosine.
Sometimes finding the coordinates of that point,
the values of sine and cosine,
can’t be done manually even with the most clever geometry.
What are the coordinates of the point on the unit circle
corresponding to the angle \(53°?\)
Note \(\cos\bigl(-\theta\bigr) = \cos\bigl(\theta\bigr)\)
and \(\sin\bigl(-\theta\bigr) = -\sin\bigl(\theta\bigr).\)
More generally, you just gotta know
the common values of sine and cosines in the first quadrant,
then translate everything from there as needed.
What are the coordinates of the point on the unit circle
corresponding to the angles
\(120°?\) or \(-45°?\) or \(210°?\)
Summarize everything with the classic “unit circle” drawing.
There are four other trig functions to know:
tangent and secant
and cotangent and cosecant.
Tangent is the only special one:
it represents the slope of the line
passing through the origin and the terminal point of \(t.\)
The others you may take to just be a notational convenience.
What is the value of
\(\tan(45°)?\) or \(\tan(30°)?\) or \(\cot(30°)?\)
or \(\sec(45°)\) or \(\csc(-60°?)\) or \(\cot(30°?)\)
\(\csc(-45°)?\) or \(\sec(210°?)\) or \(\tan(-420°?)\)
Perspective note:
the textbook presents this material differently,
from the perspective of radian measure of an angle.
On the unit circle, starting at the point \((1,0)\)
and moving around the circle counterclockwise
tracing out an arc of length \(t\)
we land at the terminal point corresponding to \(t.\)
Note that \(t\) is the radian measure of the angle
subtended by that traced arc.
Just like every angle has a principal angle
between \(0°\) and \(360°\) to which it’s coterminal,
arclength \(t\) has a reference number \(\overline{t}\)
between \(0\) and \(2\pi\) that correspond to the same terminal point.
In radian measure, the terminal point for \(t\) has coordinates
\(\bigl(\cos(t), \sin(t)\bigr)\).
Goal: commit all of this to memory.
For any of the six \(\mathtt{trig}\) functions
you should know the exact value of
\(\mathtt{trig}(\theta)\) for any angle \(\theta\)
that’s a multiple of 30° or 45°.
So memorize those values for sine and cosine in the first quadrant,
know how those values translate to other quadrants,
and know how to determine the values of the other four trig functions
from sine and cosine.
Planar Terrestrial Navigation
First, some history.
The words geometry and trigonometry
each come from the Latin
for “earth measure” and “triangle measure” respectively.
The formal study of all this mathematics was initiated ~2500 years ago
to measure land, measure the earth as a whole,
to determine the earth’s position in space and distance from the stars,
and later to study the “science of artillery” (projectile motion)
and the invention of pendulum and time keepers.
We’ll talk about the details of some of this mathematics later.
For now, let’s get acquainted with contemporary conventions
for describing the geometry of the earth’s surface.
Today we have a well established
geographic coordinate system
imposed on the surface of the earth
with lines of latitude (parallels) parallel to the equator
and lines of longitude (meridians) all passing through the poles.
The actual enumerations of the lines of latitude correspond to angles.
The latitude of a point, conventionally denoted \(\varphi,\)
is the angle ±90° of that point above/below the equator from the earth’s center,
and the longitude of a point, conventionally denoted \(\lambda,\)
is the angle ±180° of that point east/west from the prime meridian,
the meridian passing through the Royal Observatory in Greenwich.
While the earth is basically spherical and has a slight curvature,
on small scales the earth is sufficiently flat
that all the mathematics of planar geometry applies.
That is, on a small enough scale the system of
What is the area of Colorado?
is it rectangular???
Terrestrial navigation relies heavily on rectangular/polar coordinates.
Draw a
horizontal coordinate system
and label the
observer,
horizon (circle around observer),
zenith (straight above the observer, normal to the earth),
some arbitrary object in the sky,
its azimuth (\(\alpha\)) and its altitude (or altitude angle),
and use the words
elevation (or inclination)
and depression (or declination).
Distance and heading of a crow
Draw a plane from a top-down view and label its
heading (direction it’s pointing/facing)
and its track (the direction its actually going)
and its course (intended direction)
which might be different because of the wind or something.
Draw some other object and describe its
bearing relative to the plane and relative to north.
Directions are usually described as an angle
measured clockwise from north (or south).
E.g. “N 34°56′ W” refers to the direction
34°56′ rotated counterclockwise from north.
Whether “north” refers to geographic (true) north
or magnetic north or grid north depends on context.
Map making and projections: conformal maps.
A path of constant bearing
is a rhumb line is a loxodrome,
which is not a great circle, a orthodrome.
Secants of latitude and Mercator’s projection.
No map of the earth preserves all three of distance, shape, and direction
(which maps preserve 2/3 though?)
The two most commonly used global projections are
the transverse Mercator projection and the Lambert conformal conic projection.
straight projection onto a tangent plane,
straight projection onto a secant plane
cylindrical or cone projection.
That one app that let's you drag around countries.
“Right Triangle” Trigonometry
Given an angle and a side-length in a right triangle,
all the other measurements are determined by those two.
How do we calculate the other measurements?
In addition to coordinates on the unit circle
and in addition to scaling percentages of projections,
sine and cosine and tangent
can be thought of as ratios of the side-lengths
of a right triangle relative to an acute angle \(\theta.\)
(This is the way it’s classically taught in primary school.)
\[
\sin\bigl(\theta\bigr) \!=\! \frac{\text{“opposite”}}{\text{“hypotenuse”}}
\quad \cos\bigl(\theta\bigr) \!=\! \frac{\text{“adjacent”}}{\text{“hypotenuse”}}
\quad \tan\bigl(\theta\bigr) \!=\! \frac{\text{“opposite”}}{\text{“adjacent”}}
\]
Show that its the same as the “projections” interpretation.
Note cosine is literally the angle complement of sine
The other trigonometric functions
cosecant and secant and cotangent
are still the reciprocals of the original three.
\[
\csc\bigl(\theta\bigr) \!=\! \frac{\text{“hypotenuse”}}{\text{“opposite”}}
\quad \sec\bigl(\theta\bigr) \!=\! \frac{\text{“hypotenuse”}}{\text{“adjacent”}}
\quad \cot\bigl(\theta\bigr) \!=\! \frac{\text{“adjacent”}}{\text{“opposite”}}
\]
And the three Pythagorean identities still hold,
\[
\sin^2\bigl(\theta\bigr) + \cos^2\bigl(\theta\bigr) = 1
\quad \tan^2\bigl(\theta\bigr) + 1 = \sec^2\bigl(\theta\bigr)
\quad 1 + \cot^2\bigl(\theta\bigr) = \csc^2\bigl(\theta\bigr)
\]
Those “nice” output values of sine and cosine we’ve committed to memory
correspond to 30°-60°-90° and 45°-45°-90° triangles
where the side-lengths are nicely proportional.
More generally true though, the angles in any triangle must sum to 180°.
Prove that the angles in any triangle must sum to 180°.
Draw a generic triangle with angles labelled \(\alpha\) and \(\beta\) and \(\gamma\)
(and introduce the words for those Greek letters),
and sketch the line parallel to the base through the opposite vertex
and start labelling complementary and supplementary angles.
Start drawing right triangles
with a single acute angle and single side-length labelled —
of all different sizes and shapes and orientations
— and fill in the missing side-lengths.
One last fact: the area of a triangle is ½×(base)×(height)
where “height” is the height of the triangle’s altitude
perpendicular to a side chosen as the “base”.
This is because a triangle always occupies half the space
of the base×height rectangle that “bounds” it.
If we don’t know this height for a given base though,
but we do know the adjacent angle,
we can calculate the area to be ½×(base)×(slant)×\(\sin(\theta).\)
Generically \(\frac{1}{2}AB\sin(\theta)\) is the formula.
Inverse Trigonometric (Cyclometric) Functions
Given two side-lengths in a right triangle,
all the other measurements are determined by those two.
How do we calculate the other measurements?
The inverse trigonometric (or cyclometric or arcus) functions —
arccosine, arcsine, arctangent,
arcsecant, arccosecant, and arccotangent
— are the inverses of the trigonometric functions.
They each take a ratio/scaling-factor as input
and return the corresponding angle/arclength as output.
E.g. since \(\tan(45°) = 1\) we also have
\(\operatorname{arctan}\bigl(1\bigr) = 45°.\)
The notation \(\operatorname{arctan}\) and \(\tan^{-1}\)
mean the same thing, and tan-1
is how most calculators are labelled.
Draw a bunch of examples of right triangles
with only two side-lengths labelled,
and figure out the measure of the unlabelled angles.
There is a cute class of exercises
that often come along with these things.
If \(\operatorname{arctan}\Bigl(\frac{5}{2}\Bigr) = \theta\)
and \(\theta\) is an acute angle.
what must \(\sin(\theta)\) be?
If \(\operatorname{arccos}\Bigl(\frac{\sqrt{x}}{x-1}\Bigr) = \theta\)
and \(90° \leq \theta \leq 180°,\)
what must \(\tan(\theta)\) be?
When working only with right-triangles
outside the context of coordinate geometry
we don’t have to consider the domain and range of these functions.
But in the context of coordinate geometry,
there is a quirk to be aware of here.
Just like the function \(f(x) = x^2\) has no true inverse
but instead only its positive and negative branches
have inverses \(\sqrt{x}\) and \(-\sqrt{x},\)
the definition of the inverse trig functions
depends on a branch cut of their domain.
I.e. whereas \(\tan\bigl(\operatorname{arctan}(x)\bigr) = x,\)
\(\operatorname{arctan}\bigl(\tan(\theta)\bigr) \neq \theta\) in general.
Notice how the following questions are different:
What is the value of \(\operatorname{arcsin}\bigl(\frac{1}{2}\bigr)?\)
For what angles \(\theta\) does \(\sin(\theta) = \frac{1}{2}\)
Briefly, arccosine and arcsecant and arccotangent
only return angles in the upper hemisphere,
and arcsine and arccosecant and arctangent
only return angles in the right hemisphere.
This convention exists
so arcsine and arccosine and arctangent and arccotangent
are “continuous” (don’t have breaks) on their domain.
Then the range for arcsecant nearly matches the range of arccosine
and the range for arccosecant nearly matches the range of arcsine.
The Law of Sines & Law of Cosines
Given three pieces of information in a general triangle
(angle measurements and side-lengths)
all the other measurements are determined by those three.
How do we calculate the other measurements?
Give pop quiz on day two.
On the board, for reference, draw a generic triangle
with side-lengths labelled \(A\) and \(B\) and \(C\)
and opposing angles labelled
\(\alpha\) and \(\beta\) and \(\gamma\) respectively.
Demonstrate problem-solving at the end.
Everything we’ve been doing with right triangles
generalizes to general triangles,
and the vehicles of this generalization are
the law of sines and the law of cosines.
First the law of sines:
Given a triangle with side-lengths \(A\) and \(B\) and \(C\)
and opposing angles \(\alpha\) and \(\beta\) and \(\gamma\) respectively,
\[\frac{A}{\sin(\alpha)} = \frac{B}{\sin(\beta)} = \frac{C}{\sin(\gamma)}\,.\]
Given a triangle with acute angles \(\alpha\) and \(\beta\)
and opposing side-lengths \(A\) and \(B\) respectively,
drop a perpendicular from the unlabelled angle to the unlabelled side,
cutting the original triangle into two right triangles,
and name the length of this perpendicular \(X.\)
Now \(X\) is the “opposite side” of two right triangles,
one with angle \(\alpha\) and hypotenuse \(B\)
and another with angle \(\beta\) and hypotenuse \(A.\)
This indicates that \[X = B\sin(\alpha) = A\sin(\beta),\]
but \(X\) is incidental in this equation;
the equality of the other two expressions in the law of sines:
\[\frac{\sin(\alpha)}{A} = \frac{\sin(\beta)}{B} = \frac{\sin(\gamma)}{C}\,.\]
And this proof even extends to obtuse-angled triangles
by, in the illustration,
extending the base and drawing the perpendicular outside the triangle:
for the obtuse angle \(\alpha\) it’s critical to note
that \(\sin(180°-\alpha) = \sin(\alpha).\)
Draw some generic AAS triangles
with at least one pair of an opposing side/angle
and another angle and fill in the missing measurements.
(law of sines)
Draw an SSA triangle with 2′ and 3′ and 27°
where the angle between the 2′ and 3′ sides is clearly obtuse
and fill in the missing measurements.
(law of sines requiring arcsin)
But there’s a subtlety to that last example.
Our answer is correct if we trust the scale of the drawing,
but there is actually another triangle
that has the same initial measurements.
Draw the other SSA triangle with 2′ and 3′ and 27°
where the angle between the other 2′ and 3′ sides is clearly acute
and fill in the missing measurements.
(law of sines requiring arcsin,
where the target angle is 90° plus what arcsine reports)
Note that if you’re handed information
that doesn’t sincerely correspond to a triangle,
trying to solve for the remaining sides you’ll run into
a computation error from the domain of arcsine or arccosine or something.
Sketch an impossible SSA triangle
with side-lengths 2′ and 10′ and a 50° opposite the short side.
Now towards the law of cosines.
The so-called projection laws
follow from the same idea as the law of sines,
except instead of looking at the sine of those angle we look at the cosine.
If the once-unlabelled side of that same triangle is now labelled \(C,\)
the perpendicular bisector \(X\) cuts \(C\) into two lengths.
One of those lengths is \(B\cos(\alpha)\) and the other is \(A\cos(\beta),\)
to together they sum to \(C.\)
Writing this altogether
\( \displaystyle A \!=\! B\cos(\gamma) \!+\! C\cos(\beta) \)
\( \displaystyle B \!=\! A\cos(\gamma) \!+\! C\cos(\alpha) \)
\( \displaystyle C \!=\! A\cos(\beta) \!+\! B\cos(\alpha) \)
The law of cosines follows from an algebraic manipulation
of these projection laws.
Within each projection law,
first multiply through by another copy of the isolated side-length:
\[\begin{cases}
A^2 = AB\cos(\gamma) + AC\cos(\beta)
\\B^2 = AB\cos(\gamma) + BC\cos(\alpha)
\\C^2 = AC\cos(\beta) + BC\cos(\alpha)
\end{cases}\]
Then subtract any two equations from the other,
and cancelling terms we get:
\[\begin{align*}
C^2 \!-\! A^2 \!-\! B^2
&= AC\cos(\beta) \!+\! BC\cos(\alpha)
\!-\! \bigl( AB\cos(\gamma) \!+\! AC\cos(\beta) \bigr)
\!-\! \bigl( AB\cos(\gamma) \!+\! BC\cos(\alpha) \bigr)
\\ C^2 &= A^2 \!+\! B^2 \!-\! 2AB\cos(\gamma)
\end{align*}\]
Draw some generic SAS triangles
and fill in the missing measurements.
(law of cosines)
Draw a generic SSS triangles
and fill in the missing measurements.
(law of cosines requiring arccos)
The law of cosines and the \(\frac{1}{2}AB\sin(\gamma)\) formula
for the area of a triangle together lead to another area formula
strictly in terms of the triangle’s side-lengths.
Letting \(S = \frac{1}{2}(A+B+C),\)
a number referred to as the semiperimeter of the triangle,
the area of the triangle will be \(\sqrt{S(S-A)(S-B)(S-C)}.\)
This is called Heron’s formula.
The Law of Sines & Law of Cosines Redux
General results and special cases.
Example: the Pythagorean Theorem is a special case of the law of cosines.
(prove them both)
Consider two points \(P\) and \(Q\) on the boundary of a circle
centered at a point \(O.\)
For any point \(R\) (distinct from \(P\) or \(Q\)) on the the circle’s boundary,
\(2\angle PRQ = \angle POQ.\)
A cyclic quadrilateral is one whose vertices
all lie on a common circle.
In a cyclic quadrilateral,
the product of the lengths of the diagonals
will be equal to the sum
of the products of the lengths of the opposite sides.
The “converse” is also true.
The law of cosines is an established corollary to this:
wiki
If a cyclic quadrilateral’s diagonals are orthogonal,
then the ray perpendicular to any side passing through the intersection of the diagonals
will intersect the opposite side at its midpoint.
Then Brahmagupta's Theorem appears to be the analog of the law of cosines / Heron's formula for such a quadrilateral.
Let \(A\) and \(B\) and \(C\) be the vertices of a triangle,
and let \(O\) be any point internal to the triangle.
Consider the rays \(\overrightarrow{AO}\)
and \(\overrightarrow{BO}\) and \(\overrightarrow{CO},\)
and let \(A'\) and \(B'\) and \(C'\) be the points
where those rays respectively intersect
the side of the triangle opposite the vertex from which they emanate.
Then we have
\[
\frac{|BA'|}{|CA'|}
\times\frac{|CB'|}{|AB'|}
\times\frac{|AC'|}{|BC'|}
=1 \,.
\]
Review “transformations of functions”
from College Algebra class before next week.
Graphs of Trigonometric Functions
Today we’re taking this class back to College Algebra for a moment.
One of the major themes of that class
was to build a library of functions,
and to understand the geometry of their graphs.
Today we do the same,
adding all the trigonometric functions to our library
and analyzing their graphs.
Pop open Desmos and explain how the graphs of
sine, cosine, and tangent come from the unit circle,
and how the graphs of cosecant, secant, and cotangent
come from the graphs of their reciprocal counterparts.
Recall how you can sketch the graph of
\(a f\bigl(k(x-b)\bigr)+v,\)
just by knowing the graph of \(f\)
and thinking of the parameters
\(a,\) \(b,\) \(k,\) and \(v\)
as linear “transformations” of the base graph of \(f\)?
We can do the same with trigonometric functions.
Pop open Desmos with the graph of base sine
to progressively transform.
For a sinusoidal curve,
its period is the length of its minimal repeating segment,
its amplitude is the highest value it attains,
and its phase shift is it’s horizontal offset.
The graph of the base sine function
has a period of \(360°,\) amplitude of \(1,\)
and no phase shift or vertical shift
What does the graph of \(\sin(x)+3\) look like?
What about \(4\sin(x)?\)
What about \(\sin\bigl(x+90°\bigr)?\)
What about \(\sin(2x)?\)
What about \(\sin(-x)?\)
The transformed graph of \[ a\sin\big(k(x-b)\big) +v \]
will have period \(360°/k\), amplitude \(|a|\),
phase shift \(b,\) and vertical shift \(v.\)
Manually sketch a graph of the function
\( 2\sin\bigl(x-60°\bigr). \)
Manually sketch a graph of the function
\( \sin\bigl(2x-120°\bigr). \)
Manually sketch a graph of the function
\( \frac{1}{2}\cos\bigl(x-30°\bigr). \)
Manually sketch a graph of the function
\( \tan\bigl(\frac{1}{3}x\bigr)+2. \)
Manually sketch a graph of the function
\( 3\sec\bigl(x+240°\bigr). \)
Trigonometric Identities
We’re going to speed-run this topic
because it’s not that important for y’all.
First review of some identities we already know.
Cofunction identities,
and the fact that cosine is an even function
and sine is an odd function:
Then all of these extend naturally
to tangent and secant and cosecant and cotangent too.
Pythagorean identities are chill.
\(\sin^2(\theta) + \cos^2(\theta) = 1\)
\(\tan^2(\theta) + 1 = \sec^2(\theta)\)
\(1 + \cot^2(\theta) = \csc^2(\theta)\)
Sum-of-angles formulas:
It’s not prudent to have these memorized, especially not the one for tangent.
What is of value though is being able to sketch the geometric picture
to figure them out.
Then the Double-angle formulas
follow from those, letting \(\theta = \phi,\)
and half-angle formulas follow from those
together with the first Pythagorean identity.
Question: who cares?
If you’re interested in the exact forms (not just decimal approximations)
of the outputs of trigonometric functions,
the sum-of-angles, and half-angle formulas can get you beyond
just the multiples of 30° and 45°.
What is the exact value,
expressed in terms of radicals if necessary,
of \(\cos\bigl(15°\bigr)?\)
Note \(\cos\bigl(\frac{\pi}{12}\bigr) = \cos\bigl(\frac{\pi}{3} - \frac{\pi}{4}\bigr)\)
OR \(\cos\bigl(\frac{\pi}{12}\bigr) = \cos\bigl(\frac{\pi/6}{2}\bigr).\)
In general regarding who cares?
this is usually the first lesson in a much larger study
of the algebraic/analytic properties of trigonometric functions,
which this isn’t the focus of this class.
Harmonic Motion
TK
The invention of pendulum and time keepers.
Vectors in Two-Dimensional Space
Sketch the coordinate axes for positive \(x\)
and sketch the vectors \(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle\)
as running examples to use for all the following definitions.
Vectors help us talk about geometry beyond two-dimensional space.
In particular, we need them to talk about “directions”
since talking about a “slope” is no longer sufficient.
Define vector \(\bm{v} = \langle v_1,v_2 \rangle\)
and its the horizontal and vertical components.
A point and a vector carry the same data, but with a different connotation:
a point is static whereas a vector denotes movement —
like a displacement or velocity or force.
Define initial and terminal point
and the \(\overrightarrow{AB}\) notation,
and the nuance of the initial point being the origin by default.
Talk about \(\bm{v}\) versus \(\vec{v}.\)
Define vector versus scalars.
Sum and scalar multiple of vectors (with picture!)
and the resultant vector
both for sum of vectors and difference of vectors.
\[
\bm{u} = \langle u_1, u_2 \rangle
\quad
\bm{v} = \langle v_1, v_2 \rangle
\quad
\implies
\quad
\bm{u} + \bm{v} = \langle u_1+v_1, u_2+v_2 \rangle
\quad
k\bm{u} = \langle k u_1, k u_2 \rangle.
\]
Define length/magnitude of a vector
and the zero vector
and unit vectors \(\bm{\hat{v}}\) (“vee hat”)
and the unit coordinate vectors
\({\mathbf{i} = \langle 1,0 \rangle}\)
and \({\mathbf{j} = \langle 0,1 \rangle}\)
and that we sometimes write \(\bm{v} = \langle v_1, v_2\rangle\)
as \(v_1\mathbf{i} + v_2\mathbf{j}.\)
A vector’s magnitude encodes not only its direction
but the magnitude in which it is heading in that direction.
If a vector \(\bm{v}\) is an airplane’s velocity vector,
then \(|\bm{v}|\) is the plane’s speed in that direction.
Suppose an airplane has velocity vector
of \(\langle -192, 494 \rangle\) mph;
this is the plane’s heading.
What is the plane’s speed?
How do you express this heading as an angle?
Suppose an airplane has velocity vector
of \(\langle -192, 494 \rangle\) mph,
but there is a 30 mph wind from the south.
What is the plane’s speed and course?
The Dot Product & Projections
The language of vectors gives us a simpler computation
to talk about the angle between them.
Given two vectors
\(\bm{u} = \langle u_1,u_2 \rangle\)
and \(\bm{v} = \langle v_1,v_2 \rangle,\)
their dot product \(\bm{u}\cdot\bm{v}\)
(sometimes called the scalar product or inner product)
is the number defined as the result
of the calculation \({\bm{u}\cdot\bm{v} = u_1v_1+u_2v_2.}\)
Sketch the coordinate axes for positive \(x\)
and sketch the vectors \(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle\)
and compute \(\bm{u}\cdot\bm{v}.\)
Alternatively if \(\theta\) is the angle between \(\bm{u}\) and \(\bm{v},\)
then \(\bm{u}\cdot\bm{v} = |\bm{u}||\bm{v}|\cos(\theta).\)
These definitions are equivalent due to the law of cosines.
Write the law of cosines as
\({ |\bm{u} - \bm{v}|^2 = |\bm{u}|^2 + |\bm{v}|^2 - 2|\bm{u}||\bm{v}|\cos\bigl(\theta\bigr)} \)
and expand everything out in terms of components.
Calculate the measure of the angle between
\(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle.\)
Note the dot product is commutative,
\(\bm{u}\cdot\bm{v}=\bm{v}\cdot\bm{u},\)
since multiplication is commutative.
If two vectors are perpendicular (orthogonal)
then \(\bm{u}\cdot\bm{v}=0,\)
and if two vectors are parallel
then \(\bm{u}\cdot\bm{v} = |\bm{u}||\bm{v}|.\)
Intuitively then, the dot product is the product of the magnitudes of two vectors
“correcting for” how much they aren’t pointing in the same direction.
This computation plays an important role in answering the question
“how much of a vector is pointing in the direction of another vector?”
For \(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle\)
sketch the projection of \(\bm{u}\) onto \(\bm{v},\)
and sketch the projection of \(\bm{u}\) onto \(\bm{v}^\perp,\)
before defining these things and doing calculations.
For \(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle\)
compute \(\operatorname{proj}_{\bm{v}}(\bm{u}).\)
The projection of \(\bm{u}\) onto \(\bm{v}\) (a vector),
denoted \(\operatorname{proj}_{\bm{v}}(\bm{u}),\)
is calculated as
\[
\operatorname{proj}_{\bm{v}}(\bm{u})
= \bigl(|\bm{u}|\cos(\theta)\bigr)\bm{\hat{v} }
= \Bigl(\frac{\bm{u}\cdot\bm{v}}{|\bm{v}|}\Bigr)\bm{\hat{v}}
= \Bigl(\frac{\bm{u}\cdot\bm{v}}{|\bm{v}|^2}\Bigr)\bm{v}
\]
Imagine a light shines directly down, perpendicularly, onto \(\bm{v};\)
\(\operatorname{proj}_{\bm{v}}(\bm{u})\) is the “shadow” that \(\bm{u}\) casts onto \(\bm{v}.\)
The component of \(\bm{u}\) along \(\bm{v}\) (a number)
is the length of the projection of \(\bm{u}\) onto \(\bm{v},\) i.e.
\[
\bigl|\operatorname{proj}_{\bm{v}}(\bm{u})\bigr|
= \frac{\bm{u}\cdot\bm{v}}{|\bm{v}|}
= |\bm{u}|\cos(\theta)
\]
Desmos Demo for the next part.
For a vector \(\bm{v},\) let \(\bm{v}^\perp\) (“vee perp”)
denote the vector perpendicular (orthogonal) to \(\bm{v}\)
and of the same magnitude such that, following the right-hand rule,
the counterclockwise angle from \(\bm{v}\) to \(\bm{v}^\perp\) is 90°.
In terms of its components, if \(\bm{v} = \langle v_1,v_2\rangle\)
then \(\bm{v}^\perp = \langle -v_2,v_1\rangle.\)
We can resolve a vector \(\bm{u}\)
with respect to a coordinate system defined by \(\bm{v}\) and \(\bm{v}^\perp\)
by writing it as a sum of orthogonal vectors,
\[\begin{align*}
\bm{u} &= \bigl(|\bm{u}|\cos(\theta)\bigr)\bm{\hat{v}} + \bigl(|\bm{u}|\sin(\theta)\bigr)\bm{\hat{v}}^\perp
\\ &= \Bigl(\operatorname{proj}_{\bm{v}}(\bm{u})\Bigr) + \Bigl(\operatorname{proj}_{\bm{v}^\perp}(\bm{u})\Bigr)
\\ &= \Bigl(\operatorname{proj}_{\bm{v}}(\bm{u})\Bigr) + \Bigl(\bm{u} - \operatorname{proj}_{\bm{v}}(\bm{u})\Bigr)
\,.
\end{align*}\]
For \(\bm{u} = \langle 3,4\rangle\)
and \(\bm{v} = \langle 5,-2\rangle\)
compute \(\operatorname{proj}_{\bm{v}^\perp}(\bm{u}).\)
Three-Dimensional Rectangular Geometry
In three-dimensional space, we add a \(z\)-axis
perpendicular (orthogonal) to the usual \(x\)- and \(y\)-axis
to create a rectangular coordinate system called \(xyz\)-space.
The \(z\)-axis is oriented relative to the \(xy\)-plane
in accordance to the right-hand rule.
Any pair of axes span a coordinate plane,
and these coordinate planes split \(xyz\)-space into eight octants.
Each point can be described by a ordered triple \((x,y,z).\)
Plot the points
\((-1,0,0)\) and \((0,3,0)\) and \((0,0,4)\)
and \((1,2,3)\) and \((-1,4,-5)\)
Given two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2),\)
the distance between them is \( \sqrt{(x_2\!-\!x_1)^2+(y_2\!-\!y_1)^2+(z_2\!-\!z_1)^2}, \)
and their midpoint has coordinates \((\overline{x}, \overline{y}, \overline{z})\)
given by the formulas
\( \overline{x} = \frac{1}{2}(x_1\!+\!x_2) \)
and \( \overline{y} = \frac{1}{2}(y_1\!+\!y_2) \)
and \( \overline{z} = \frac{1}{2}(z_1\!+\!z_2).\)
What’s the distance between the points
\((1,2,3)\) and \((-1,4,-5)?\)
What’s the midpoint between the points
\((1,2,3)\) and \((-1,4,-5)?\)
What’s the distance between the point
\((1,2,3)\) and the generic point \((x,y,z)?\)
What’s an equation that represents all points \((x,y,z)\)
such that the distance between
\((1,2,3)\) and \((x,y,z)\) is seven?
A sphere of radius \(r\) centered at \((h,k,\ell)\)
has equation \( r^2 = (x-h)^2+(y-k)^2+(z-\ell)^2\,. \)
In three-dimensional space,
two points uniquely determine a line
and three non-colinear points uniquely determine a plane.
Generically, a single linear equation determines a plane.
More generally, a single equation will carve out
a locally two-dimensional surface in space.
Locally one-dimensional regions in space, curves,
and specifically lines,
must now be viewed as the intersection of two surfaces,
and so can only be described by two equations.
Anyways.
Sketch the planes
\(x=4\) and \(y=-2\) and \(z=7\) and \(x=z.\)
If a sphere intersects a plane in space,
the resulting shape of their intersection, generically a circle,
is called the trace of the sphere in the plane.
What is the radius of the trace of the sphere
\( 7^2 = (x-1)^2+(y-2)^2+(z-3)^2 \)
in the plane \(x=4?\)
Vectors in Three-Dimensional Space
Vectors in \(xyz\)-space have three components.
The magnitude of \(\bm{v} = \langle v_1, v_2, v_3\rangle\) is calculated as
\(|\bm{v}| = \sqrt{v_1^2+v_2^2+v_3^2}\,.\)
In addition to the unit coordinate vectors
\({\mathbf{i} = \langle 1,0,0 \rangle}\)
and \({\mathbf{j} = \langle 0,1,0 \rangle,}\)
we now have \({\mathbf{k} = \langle 0,0,1 \rangle}.\)
For vectors \({\bm{u} = \langle u_1,u_2,u_3 \rangle}\)
and \({\bm{v} = \langle v_1,v_2,v_3 \rangle}\)
their dot product is calculated as
\({\bm{u}\cdot\bm{v} = u_1v_1 + u_2v_2 + u_3v_3} = |\bm{u}||\bm{v}|\cos(\theta).\)
Let \(\bm{u} = \langle 1,2,3 \rangle\)
and \(\bm{v} = \langle -1,4,-5 \rangle.\)
Plot them.
What is their sum?
What are their lengths?
What are the unit vectors in their directions?
What is their dot product?
What is the angle between these vectors?
What is the area of the triangle they frame?
What are the projections of one onto the other?
The direction angles of a vector \(\bm{v}\)
are the three angles between \(\bm{v}\) and each of the three coordinate axes.
The cosines of these angles are referred to as the vector’s direction cosines.
If \(\alpha\) and \(\beta\) and \(\gamma\) are the direction angles of \(\bm{v}\)
with the \(x\)- and \(y\)- and \(z\)-axis respectively, then
\[
\cos(\alpha) = \frac{v_1}{|\bm{v}|}
\quad \cos(\beta) = \frac{v_2}{|\bm{v}|}
\quad \cos(\gamma) = \frac{v_2}{|\bm{v}|}
\qquad \bm{v} = \Bigl\langle |\bm{v}|\cos(\alpha), |\bm{v}|\cos(\beta), |\bm{v}|\cos(\gamma) \Bigr\rangle \,.
\]
Let \(\bm{u} = \langle 1,2,3 \rangle.\)
What are the direction angles of \(\bm{v}?\)
Express \(\bm{v}\) in terms of its direction cosines.
The Cross-Product & Areas
Give pop quiz on day two.
Give two non-parallel vectors in space,
there must be a unique direction
that is normal/orthogonal to both of them
… but how do we actually compute it?
Today we are going to be the determinant of a matrix,
but it’s not important that you know those terms,
only recognize them in case they come up in your life again.
There are quite a few “algorithms” for computing determinants.
We’ll only cover one in class,
the rule of Sarrus,
but if you find another online you like better,
that is certainly fine.
Compute the determinant of the following \(3 \times 3\) matrices.
Generically the resulting vector will be orthogonal to both \(\bm{u}\) and \(\bm{v}\)
and will point in the direction indicated by the right-hand rule.
It’s length will be proportional to the sine of the angle between \(\bm{u}\) and \(\bm{v}.\)
Specifically, the magnitude of the cross product will be
the product of the vectors’ individual magnitudes
weighted by the sine of the angle between them:
\(|\bm{u} \times \bm{v}| = |\bm{u}||\bm{v}|\sin(\theta).\)
Not coincidentally, this number also equals the area of the parallelogram
(twice the area of the triangle) framed by \(\bm{u}\) and \(\bm{v}\).
As a special case, two vectors \(\bm{u}\) and \(\bm{v}\) are parallel
if and only if \(\bm{u} \times \bm{v} = \bm{0}.\)
What is the area of the triangle in space
with vertices located at the coordinates
\((1,2,3)\) and \((-1,4,-5)\) and \((2,-2,3)?\)
Given a plane in three-dimensional space
a vector \(\bm{n}\) is normal to the plane
if it is orthogonal to every vector parallel to the plane;
i.e. \(\bm{n}\cdot\bm{v}=0\)
for every vector \(\bm{v}\) parallel to the plane.
Three non-colinear points in space uniquely determine a plane.
Given three such points \(P\) and \(Q\) and \(R\)
we can calculate a vector normal to that plane as
\(\overrightarrow{PQ}\times\overrightarrow{PR}.\)
What is a vector normal to the plane
containing the three points
\((-1,4,-5)\) and \((2,3,1)\) and \((1,-4,1)?\)
The scalar triple product
of vectors \(\bm{u}\) and \(\bm{v}\) and \(\bm{w}\)
is calculated as
\[
\bm{u} \cdot \bigl(\bm{v} \times \bm{w}\bigr)
= \operatorname{det}\!\begin{pmatrix}
u_1 & u_2 & u_3
\\ v_1 & v_2 & v_3
\\ w_1 & w_2 & w_3
\end{pmatrix}.
\]
and gives the volume of the parallelepiped,
the three-dimensional analog of a parallelogram,
(and twice the volume of the tetrahedron)
framed by those three vectors.
Specifically, the vectors \(\bm{u}\) and \(\bm{v}\) and \(\bm{w}\)
all lie in the same plane (are coplanar)
if and only if \(\bm{u}\cdot\bigl(\bm{v}\times\bm{w}\bigr)=0.\)
What is the volume of the tetrahedron
with vertices located at the points with coordinates
\((-1,4,-5)\) and \((2,3,1)\) and \((1,-4,1)\) and \((1,3,-1)?\)
Lines and Planes in Three-Dimensional Space
What is the “algebra” underlying a line or plane in space?
Just like in previous math classes we spent a lot of time
studying lines and linear equation in the \(xy\)-plane,
we’ll do the same in this class,
but now there are two different linear geometric features:
1-dimensional lines and 2-dimensional planes.
A plane doesn’t have a single slope anymore;
it depends on what direction you’re headed.
Now though we characterize a plane by its normal vector.
This is not new though:
Consider the standard form
of the line \(y = \frac{3}{4}x+\frac{15}{4},\)
which is \(4y-3x=15.\)
A plane in space passing through a point \(P = (x_0, y_0, z_0)\)
with normal vector \(\bm{n} = \langle A, B, C \rangle\)
will be the set of all points \((x,y,z)\)
such that \(\bm{n} \cdot \bigl(\langle x,y,z \rangle-P\bigr) = 0.\)
Expanding this we get the equation
\({A(x-x_0) + B(y-y_0) + C(z-z_0) = 0,}\)
which can be written as \({Ax + By + Cz = D.}\)
What’s an equation of the plane
with normal vector \(\langle 3,2,-7\rangle\)
that contains the point \((4,-5,1)?\)
Write the equation as \({Ax + By + Cz = D.}\)
What are other points on this plane?
What’s an equation of the plane
containing the three points
\((-1,4,-5)\) and \((2,3,1)\) and \((1,-4,1)?\)
A line in three dimensional space must be described
either as the intersection of two planes
or as a “vector equation” of a parameter \(t.\)
The line passing through a point \(P = (x_0,y_0,z_0)\)
in the direction of the vector \(\bm{v} = \langle v_1,v_2,v_3 \rangle\)
can be described by the parametric equation
\[P + \bm{v}t = \bigl\langle x_0+v_1t, y_0+v_2t, z_0+v_3t \bigr\rangle.\]
What’s an equation of the line
that passes through the points \((0,-3,7)\) and \((1,-1,4)?\)
Does it matter which starting point we use?
It doesn't matter, but just changes the point
a value of \(t\) corresponds to.
Two non-parallel planes in space will intersect along a line.
If the planes have normal vectors \(\bm{n}_1\) and \(\bm{n}_2\),
then the direction vector of the line will be \(\bm{n}_1 \times \bm{n}_2,\)
and any solution to the system of equations consisting of the planes
will result in a point on the line.
What is a parameterization of the line along which
the planes \(2x-4y+z=1\) and \(5x-3y-z=2\) intersect?
A plane in space and a line not parallel to that plane must intersect it at a point.
For the plane \(\bm{n} \cdot \bigl(\langle x,y,z \rangle-P\bigr) = 0\)
and line \({\bigl\langle x_0 + v_1t, y_0 + v_2t, z_0 + v_3t \bigr\rangle}\)
letting \({x = x_0 + v_1t}\) and \({y = y_0 + v_2t}\) and \({z = z+0 + v_3t}\)
in the equation of the plane and solving will result in the value of \(t\)
at which the line hits the plane, which,
when plugged back into the equation of the line itself
will result in the coordinates of the intersection point.
What is the point at which the line
\(\bigl\langle 2,3,-1\bigr\rangle t + (4,1,0)\)
intersects the plane \(2x-4y+z=1?\)
Distances Between Points, Lines, and Planes
Given two geometric objects — points, lines, planes, etc —
how do we compute the (shortest) distance between them?
Compute the shortest distance to a plane from a point not on that plane.
This idea of projecting ends up being the key.
Given a plane in space with normal vector \(\bm{n}\) and containing a point \(Q,\)
and given a point \(P\) not on that plane,
if \(\overrightarrow{QP} = \bm{v},\)
the shortest distance from the point \(P\) to the plane will be
\(\bigl|\operatorname{proj}_{\bm{n}}(\bm{v})\bigr|.\)
What’s the shortest distance from the point \(\bigl(3,-3,10\bigr)\)
to the plane \({2x-3y+6z=-23?}\)
What are the coordinates of the point on that plane closest to that point?
Note \((-1,1,-3)\) lies on the plane.
The distance is \(14\) and the point is \((-1,3,-2).\)
When devising “nice” examples,
start with a Pythagorean quadruple \(\bigr(v_1, v_2, v_3, d\bigl)\)
and a non-nice point \(R\) in space.
For any permutation \(\sigma\) and constant \(c\)
travel away from \(R\) along a vector
\({\bm{n} = c\bigr\langle \pm v_{\sigma 1}, \pm v_{\sigma 2}, \pm v_{\sigma 3}\bigl\rangle}\)
with terminal point \(P\)
and build the equation of the plane passing through \(R\)
with normal vector \(\bm{n}.\)
If the length needs to be a specific number
find a Pythagorean quadruple with diagonal
a divisor of that number, and scale it up.
Compute the shortest distance to a line from a point not on that line.
Call the point \(P,\) and let \(R\) be the point on the line closest to \(P.\)
Let \(\bm{v}\) be the direction vector of the line
and let \(Q\) be the coordinates of any point on the line,
both of which can be inferred from an parameterization of the line.
Let \(\theta\) be the angle between \(\bm{u}\) and \(\bm{v}.\)
which can be computed as
\( \theta = \operatorname{arccos}\bigl(\frac{\bm{u}\cdot\bm{v}}{|\bm{u}||\bm{v}|}\bigr)\,. \)
Considering the right triangle \(\triangle PQR,\)
the distance from \(P\) to the plane —
the distance between \(P\) to \(R\) —
will be \(|\bm{u}\sin\bigl(\theta\bigr)|.\)
What’s the shortest distance from the point \(\bigl(1,-1,-1\bigr)\)
to the line \(\bigl\langle 3,1,4 \bigr\rangle t + \bigl(6,6,0\bigr)?\)
What are the coordinates of the point on that line
closest to the point?
It's \(7,\) and the closest point is \((3,5,-4).\)
Devising nice examples can be done similarly to the case
of “distance between points and planes” problems,
but instead any line in the plane
through \(R\) needs to be determined.
Compute the shortest distance between two skew (non-intersecting) lines.
Let \(P + \bm{w}t\) and \(Q + \bm{v}t\)
be parameterizations of the two lines.
If the lines are parallel,
i.e. \(\bm{w}\) and \(\bm{v}\) are parallel,
then the distance between the lines will be
the distance from the point \(P\) to the line \(Q + \bm{v}t.\)
Otherwise for each line,
there is a unique plane containing that line
that doesn’t intersect the other line.
I.e. there is a pair of parallel planes containing those lines.
Each plane will have normal vector \(\bm{n} = \bm{w}\times\bm{v},\)
and the distance between the lines
will be the distance from \(Q\) to the plane containing \(P\)
(or vice-versa).
What’s the shortest distance between the lines
\({\bigl\langle 0,-2,1 \bigr\rangle t + \bigl(3,0,2\bigr)}\)
and \({\bigl\langle 4,3,-2 \bigr\rangle t + \bigl(0,3,11\bigr)}?\)
It’s \(9.\)
For each of these types of problems,
let \(R\) denote the point on the line or plane closest to \(P.\)
All of these problems could also be solved
by finding the coordinates of the \(R\)
as the intersection of some line and plane
and computing the length of \(\overline{PR}.\)
Famously, Eratosthenes
is credited with the first meaningful attempt
at measuring the size of the earth.
He knew that on a certain day in the city of Syene
the sun would be perfectly overhead,
so he measured the length of the shadow of a
Gnomon
(sundial blade) in Alexandria about 523 miles north of the city
to determine that the central angle between the cities was about 7°.
So if that 523 miles was 7° of the circle,
then the full circumference of the earth must be 26,897 miles,
and the radius must be 4280 miles.
This is pretty close to the modern accepted average of 3963 miles.
Epicycloids and hypocucloids - paths of planets relative to the earth.
Earth-Centered-Earth-Fixed (ECEF) coordinates
Conventional Terrestrial System (CTS)
International Reference Pole (IRP)
Polar motion (rotational wobble) and the 430 day Chandler period
Conventional International Origin (CIO) of z-axis to ignore polar wobble
Difference between geocentric latitude and geodetic latitude and astronomic latitude:
astro follow gravity, direction of a plumb bob; the plumb line
geocentric uses the center of the earth;
note the plumb line doesn't point towards the center of the earth
geodetic uses the perpendicular line to the ellipsoidal surface of the earth.
Show gravity map why plumb line is screwy anyways.
The earth is an ellipsoid (oblate spheroid).
WSG 84
plot a circle vs the earth ellipsoid in desmos
Tidal force (bulge) from the moon.
"Geodetic distance"
Universal Transverse Mercator (UTM) system
