PC - Trigonometric Functions Lesson

Trigonometric Functions

Angles can be measured in degrees or in radians, but in calculus we primarily use radians to measure angles. View the video below to refresh your memory about radian measure and the relationship of angles expressed in degrees and in radians.

There are six trigonometric functions: sine θ (sin θ) , cosine θ (cos θ), tangent θ (tan θ), cotangent θ (cot θ), secant θ (sec θ), and cosecant θ (csc θ). Watch the video below to review the definitions of these functions.

Trigonometric Values at $LaTeX: 0,\:\frac{\Pi}{6},\:\frac{\Pi}{4},\:\frac{\Pi}{3},\:\frac{\Pi}{2}$ $0,\:\frac{\Pi}{6},\:\frac{\Pi}{4},\:\frac{\Pi}{3},\:\frac{\Pi}{2}$ , and their Multiples

Click HERE Links to an external site. to investigate sine, cosine, and tangent and how they relate to their graphs Links to an external site.! It may be helpful to see a table of values of the six trigonometric functions for special angles.

image of a table with the 6 trigonometric functions

Periodicity

image of sea waves A function with values that are repeated for all integral multiples of a constant increment of the independent variable is periodic. Stated another way, a function f(x) is periodic if there is a positive number p such that f(x + p) = f(x) for every value of x. You have already seen this repetition of trigonometric function values on the unit circle and in their graphs.

Reciprocal, Quotient, and Pythagorean Identities

Special identities involving reciprocal, quotient, and Pythagorean relationships are important tools for calculus. View the video below that defines and derives these identities.

Trigonometric and Inverse Trigonometric Graphs

Click HERE to view the video below to refresh your memory of the basic graphs of the six trigonometric functions. Links to an external site.

Click HERE to view the graphs of the six related inverse trigonometric functions are discussed below. Links to an external site.

Trigonometric Functions Practice

For what values of θ for equation image indicator are function values equal and what are those values?

Problem 1: $LaTeX: \sin\:\theta\:and\:\cos\theta$ $\sin\:\theta\:and\:\cos\theta$

Solution 1: $LaTeX: \sin0=\cos\frac{\pi}{2}=0\\ \sin \frac{\pi}{2}=\cos0=1\\ \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}\\ \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt[]{3} }{2}\\ \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt[]{2} }{2}\\$ $\sin0=\cos\frac{\pi}{2}=0\\ \sin \frac{\pi}{2}=\cos0=1\\ \sin\frac{\pi}{6}=\cos\frac{\pi}{3}=\frac{1}{2}\\ \sin\frac{\pi}{3}=\cos\frac{\pi}{6}=\frac{\sqrt[]{3} }{2}\\ \sin\frac{\pi}{4}=\cos\frac{\pi}{4}=\frac{\sqrt[]{2} }{2}\\$

Problem 2: $LaTeX: \tan\theta\:and\:\cot\theta$ $\tan\theta\:and\:\cot\theta$

Solution 2: $LaTeX: \tan0=\cot\frac{\pi}{2}=0\\ \tan\frac{\pi}{2}=\cot0=undefined\\ \tan\frac{\pi}{6}=\cot\frac{\pi}{3}=\frac{\sqrt[]{3}}{3}\\ \tan\frac{\pi}{3}=\cot\frac{\pi}{6}=\sqrt[]{3}\\ \tan\frac{\pi}{4}=\cot\frac{\pi}{4}=1$ $\tan0=\cot\frac{\pi}{2}=0\\ \tan\frac{\pi}{2}=\cot0=undefined\\ \tan\frac{\pi}{6}=\cot\frac{\pi}{3}=\frac{\sqrt[]{3}}{3}\\ \tan\frac{\pi}{3}=\cot\frac{\pi}{6}=\sqrt[]{3}\\ \tan\frac{\pi}{4}=\cot\frac{\pi}{4}=1$

Problem 3: $LaTeX: \sec\theta\:and\:\csc\theta$ $\sec\theta\:and\:\csc\theta$

Solution 3: $LaTeX: \sec0=\csc\frac{\pi}{2}=0\\ \sec\frac{\pi}{2}=\csc0=undefined\\ \sec\frac{\pi}{6}=\csc\frac{\pi}{3}=\frac{2}{\sqrt[]{3}}\\ \sec\frac{\pi}{3}=\csc\frac{\pi}{6}=2\\ \sec\frac{\pi}{4}=\csc\frac{\pi}{4}=\sqrt[]{2}$ $\sec0=\csc\frac{\pi}{2}=0\\ \sec\frac{\pi}{2}=\csc0=undefined\\ \sec\frac{\pi}{6}=\csc\frac{\pi}{3}=\frac{2}{\sqrt[]{3}}\\ \sec\frac{\pi}{3}=\csc\frac{\pi}{6}=2\\ \sec\frac{\pi}{4}=\csc\frac{\pi}{4}=\sqrt[]{2}$

Trigonometric Functions: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of the trigonometric and inverse trigonometric functions and their graphs.

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