The Other Trigonometric Functions: Learn It 5

Alternate Forms of the Pythagorean Identity

We can use these fundamental identities to derive alternative forms of the Pythagorean Identity, [latex]{\cos }^{2}t+{\sin }^{2}t=1[/latex]. One form is obtained by dividing both sides by [latex]{\cos }^{2}t:[/latex]

[latex]\begin{gathered}\frac{{\cos }^{2}t}{{\cos }^{2}t}+\frac{{\sin }^{2}t}{{\cos }^{2}t}=\frac{1}{{\cos }^{2}t}\\ \\ 1+{\tan }^{2}t={\sec }^{2}t\end{gathered}[/latex]

The other form is obtained by dividing both sides by [latex]{\sin }^{2}t:[/latex]

[latex]\begin{gathered}\frac{{\cos }^{2}t}{{\sin }^{2}t}+\frac{{\sin }^{2}t}{{\sin }^{2}t}=\frac{1}{{\sin }^{2}t}\\ \\ {\cot }^{2}t+1={\csc }^{2}t\end{gathered}[/latex]

alternate Pythagorean identities

[latex]1+{\tan }^{2}t={\sec }^{2}t[/latex]

[latex]{\cot }^{2}t+1={\csc }^{2}t[/latex]

If [latex]\text{cos}\left(t\right)=\frac{12}{13}[/latex] and [latex]t[/latex] is in quadrant IV, find the values of the other five trigonometric functions.
Graph of circle with angle of t inscribed. Point of (12/13, y) is at intersection of terminal side of angle and edge of circle.

periodic functions

A function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic. For the four trigonometric functions, sine, cosine, cosecant and secant, a revolution of one circle, or [latex]2\pi[/latex], will result in the same outputs for these functions. And for tangent and cotangent, only a half a revolution will result in the same outputs.swer]

[latex]\begin{gathered}\sin t=y=-\frac{\sqrt{3}}{2}\\ \cos t=x=-\frac{1}{2}\\ \tan t=\frac{\sin t}{\cos t}=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\sqrt{3}\\ \sec t=\frac{1}{\cos t}=\frac{1}{-\frac{1}{2}}=-2\\ \csc t=\frac{1}{\sin t}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2\sqrt{3}}{3}\\ \cot t=\frac{1}{\tan t}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\end{gathered}[/latex]

If [latex]\sin \left(t\right)=-\frac{\sqrt{3}}{2}[/latex] and [latex]\text{cos}\left(t\right)=\frac{1}{2}[/latex], find [latex]\text{sec}\left(t\right),\text{csc}\left(t\right),\text{tan}\left(t\right),\text{ cot}\left(t\right)[/latex].