Course Trivium
for Trigonometry & Analytic Geometry
Enumerated here are the calculations and tasks
that should become routine to students in this course.
Each is stated to have a clear goal to suggest an exercise to perform.
It is worth a student’s time to become proficient at doing these tasks
so as to not be impeded thinking about more interesting mathematics.
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Calculate the straight-line distance
between two points in space given their coordinates.
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Calculate the midpoint of a line segment
given the coordinates of its endpoints.
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Determine an equation of a line
given either the coordinates of two points it contains
or a slope and the coordinates of one point it contains,
and vice versa: recognize an equation as the equation of a line
and determine its slope.
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Determine the equation of a circle
given a radius and the coordinates of its center,
and vice versa: recognize an equation as the equation of a circle
and determine its center and radius.
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Determine the coordinates of the point
at which two given intersecting lines intersect.
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Express the measure of an angle in radians,
in degrees as a decimal (DD),
or in degrees/minutes/seconds (DMS),
given one expression for the measure of the angle.
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Calculate the principal polar coordinates of a point
given its rectangular coordinates, and vice versa.
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Calculate the angle at which a line is inclined
from a coordinate axis given its equation.
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Know an exact algebraic representation for the result
of evaluating any trigonometric function at any integer multiple
of 30° (\(\frac{\pi}{6}\)) or 45° (\(\frac{\pi}{4}\)).
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For a right triangle,
given either the lengths of two sides
or the length of one side and measure of one angle,
compute the lengths of the remaining sides
the measures of the remaining angles,
and the area of the triangle.
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For a general triangle,
given either the lengths of three sides,
the lengths of two sides and the measure of an angle,
or the lengths of one side and the measure of two angles,
determine all possible triangles that have the measurements given,
and compute the possible lengths of the remaining sides,
the measures of the remaining angles,
and the area of the triangle.
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Accurately sketch the graph of a composite
of a trigonometric function with a combination of linear functions,
and vice versa: given the graph of a composite
of a trigonometric function with a combination of linear functions,
recognize it as the graph of such a function
and determine its formula.
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Given the sines and cosines of two angles,
compute the sine and cosine of the sum and difference of those angles.
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Given the cosines of an angle,
compute the sine and cosine of half that angle.
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Calculate and sketch any linear combination of two vectors.
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Calculate the measure of the angle between two vectors.
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Determine the components of the vector
that results from projecting one vector onto another.
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Determine the components of a vector that is orthogonal
to two given vectors in three-dimensional space.
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Determine a vector parameterization of the line that contains two points
given the coordinates of those points.
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Determine an equation for the plane that contains
three non-colinear points given their coordinates.
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Determine a vector parameterization of the line
along which two intersecting planes intersect.
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Determine the coordinates of the point at which a line intersects a plane.
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Compute the shortest distance to a plane from a point.
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Compute the shortest distance to a line from a point.
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Compute the shortest distance between two skew lines.