# Trigonometric Functions on Right Triangles - Video Tutorials & Practice Problems

### Introduction to Trigonometric Functions

Given the right triangle below, evaluate $\cos\theta$.

$\cos\theta=\frac{8}{17}$

$\cos\theta=\frac{8}{15}$

$\cos\theta=\frac{15}{17}$

$\cos\theta=\frac{15}{8}$

Given the right triangle below, evaluate $\tan\theta$.

$\tan\theta=\frac35$

$\tan\theta=\frac45$

$\tan\theta=\frac43$

$\tan\theta=\frac34$

### Fundamental Trigonometric Identities

If $\tan\theta=\frac{12}{5}$, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{5},\csc\theta=\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=\frac{5}{12},\sec\theta=\frac{13}{12},\csc\theta=\frac{13}{5}$

$\sin\theta=\frac{12}{13},\cos\theta=\frac{5}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{5},\csc\theta=-\frac{13}{12}$

$\sin\theta=\frac{5}{13},\cos\theta=\frac{12}{13},\cot\theta=-\frac{5}{12},\sec\theta=-\frac{13}{12},\csc\theta=-\frac{13}{5}$

If $\sin\theta=\frac{\sqrt{17}}{17}$, find the values of the five other trigonometric functions. Rationalize the denominators if necessary.

$\cos\theta=\frac{\sqrt{17}}{4},\tan\theta=\frac14,\cot\theta=4,\sec\theta=\sqrt{17},\csc\theta=\frac{4\sqrt{17}}{17}$

$\cos\theta=\frac{\sqrt{17}}{4},\tan\theta=-\frac14,\cot\theta=-4,\sec\theta=\sqrt{17},\csc\theta=\frac{4\sqrt{17}}{17}$

$\cos\theta=\frac{4\sqrt{17}}{17},\tan\theta=-\frac14,\cot\theta=-4,\sec\theta=\frac{\sqrt{17}}{4},\csc\theta=\sqrt{17}$

$\cos\theta=\frac{4\sqrt{17}}{17},\tan\theta=\frac14,\cot\theta=4,\sec\theta=\frac{\sqrt{17}}{4},\csc\theta=\sqrt{17}$

### Introduction to Inverse Trig Functions

Given the right triangle below, use the sine function to write a trigonometric expression for the missing angle $\theta$.

$\theta=\sin^{-1}\left(\frac{5}{13}\right)$

$\theta=\sin^{-1}\left(\frac{12}{13}\right)$

$\theta=\sin^{-1}\left(\frac{5}{12}\right)$

$\theta=\sin^{-1}\left(\frac{13}{12}\right)$

Given the right triangle below, use the cosine function to write a trigonometric expression for the missing angle $\phi$.

$\phi=\cos^{-1}\left(\frac{12}{13}\right)$

$\phi=\cos^{-1}\left(\frac{13}{5}\right)$

$\phi=\cos^{-1}\left(\frac{13}{12}\right)$

$\phi=\cos^{-1}\left(\frac{5}{13}\right)$

### How to Use a Calculator for Trig Functions

What is a positive value of A in the interval $\left[0\degree,90\degree\right)$ that will make the following statement true? Express the answer in four decimal places.

$\sin A=0.9235$

$22.5568°$

$67.4432°$

$22.4432°$

$33.5438°$

What is the positive value of P in the interval $\left[0\degree,90\degree\right)$ that will make the following statement true? Express the answer in four decimal places.

$\cot P=5.2371$

$55.8102°$

$34.1898°$

$10.8102°$

$79.1898°$

What is the positive value of $D$ in the interval $\left[0,\frac{\pi}{2}\right)$ that will make the following statement true? Express the answer in four decimal places.

$\sec D=3.2842$

$0.3094$ rad

$1.2614$ rad

$0.4760$ rad

$1.0934$ rad