Enumerated here are some of the foundational tasks, mostly banal calculations, that students should know how to perform for this course. The statement of each task immediately suggests an exercise to practice. It is worth a student’s time to become proficient at doing these tasks, to be able to perform them without thought, without hesitation, so as to not impeded thinking about more interesting things.
- Calculate the distance between two points given their coordinates.
- Calculate the midpoint of a line segment given the coordinates of its endpoints.
- 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.
- 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.
- Determine the coordinates of the point at which two given intersecting lines intersect.
- Compute the distance to a line from a point.
- Express the measure of an angle in radians, in degrees as a decimal, or in degrees/minutes/seconds (DMS), given one expression for the measure of the angle.
- Calculate the polar coordinates of a point given its rectangular coordinates, and vice versa.
- Calculate the angle at which a line is inclined from a coordinate axis given its equation.
- Know an exact algebraic representation for the result of evaluating any trigonometric function at any integer multiple of \(\frac{\pi}{6}\) (30°) or \(\frac{\pi}{4}\) (45°).
- 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.
- 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 for those possibilities compute the lengths of the remaining sides, the measures of the remaining angles, and the area of the triangle.
- 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.
- Given the sines and cosines of two angles, compute the sine and cosine of the sum and difference of those angles.
- Given the cosines of an angle, compute the sine and cosine of half that angle.
- Calculate and sketch any linear combination of two vectors.
- Calculate the measure of the angle between two vectors.
- Determine the vector that results from projecting one vector onto another.
- Determine a vector that is orthogonal to two given vectors in three-dimensional space.
- Determine a vector parameterization of the line that contains two points given the coordinates of those points.
- Determine an equation for the plane that contains three non-colinear points given their coordinates.
- Determine a vector parameterization of the line along which two intersecting planes intersect.
- Determine the coordinates of the point at which a line intersects a plane.
- Compute the distance to a plane from a point.
- Compute the distance to a line from a point.
- Compute the shortest distance between two skew lines.