Trigonometric Functions

Every function so far has taken a number and returned a number. This chapter starts somewhere else: with an angle, and with the observation that a point carried around a circle returns to where it began. Measuring that rotation in radians — arc length divided by radius, a pure ratio rather than an arbitrary 360th of a turn — makes the connection between angle and distance direct, and turns questions about rotating wheels and satellites into arithmetic. Placing the angle at the centre of a unit circle then gives sine and cosine their definitions as the coordinates of a point, which is why they repeat, why they never leave $[-1,1]$, and why $\cos^2t+\sin^2t=1$ is just the Pythagorean Theorem. The remaining four functions follow as ratios of those two, and the chapter closes by reading the same definitions off a right triangle, where they become the tool for measuring a height you cannot reach.

Sections

• Angles — drawing angles in standard position, converting between degrees and radians, finding coterminal and reference angles, arc length and the area of a sector, and linear and angular speed on a circular path.
• Unit Circle: Sine and Cosine Functions — defining sine and cosine from the coordinates of a point on the unit circle, the Pythagorean Identity, the exact values at $30^\circ$, $45^\circ$, and $60^\circ$, the domain and range of both functions, and using reference angles to evaluate them anywhere.
• The Other Trigonometric Functions — tangent, secant, cosecant, and cotangent as ratios of sine and cosine, their exact values at the special angles, the even and odd properties, the fundamental identities and the alternate forms of the Pythagorean Identity, and evaluating them with a calculator.
• Right Triangle Trigonometry — the six functions as ratios of the sides of a right triangle, the special-angle values read off $30$–$60$–$90$ and $45$–$45$–$90$ triangles, cofunctions of complementary angles, and solving applied problems with angles of elevation and depression.

The sections above are adapted from Precalculus 2e, Chapter 5 by Jay Abramson, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org.