4.5 Other Trigonometric Functions and Identities

Introduction

Ratios in right triangles

Properties of the secant, cosecant, and cotangent functions

Consider a right triangle with hypotenuse of length $61$ and one leg of length $11$. Let $\alpha$ be the angle opposite the side of length $11$. Find the exact length of the other leg and then determine the value of each of the six trigonometric functions evaluated at $\alpha$. In addition, what are the exact and approximate measures of the two non-right angles in the triangle?

Complete the tables below to determine the exact values of the cosecant function at the special points on the unit circle. Enter “$u$” for any value at which $q(t) = \csc(t)$ is undefined.

$t$ | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\frac{2\pi}{3}$ | $\frac{3\pi}{4}$ | $\frac{5\pi}{6}$ | $\pi$

$\sin(t)$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$

$\csc(t)$ |

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In which quadrants is $q(t) = \csc(t)$ positive? negative?

At what $t$-values does $q(t) = \csc(t)$ have a vertical asymptote? Why?

What is the domain of the cosecant function? What is its range?

Sketch an accurate, labeled graph of $q(t) = \csc(t)$ on the figure given at the start of this activity, including the special points that come from the unit circle.

What is the period of the cosecant function?

Complete the following tables to determine the exact values of the cotangent function at the special points on the unit circle. Enter “u” for any value at which $r(t) = \cot(t)$ is undefined.

$t$ | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\frac{2\pi}{3}$ | $\frac{3\pi}{4}$ | $\frac{5\pi}{6}$ | $\pi$

$\sin(t)$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$

$\cos(t)$ | $1$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{1}{2}$ | $0$ | $-\frac{1}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-1$

$\tan(t)$ | $0$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\frac{3}{\sqrt{3}}$ | u | $-\frac{3}{\sqrt{3}}$ | $-1$ | $-\frac{1}{\sqrt{3}}$ | $0$

$\cot(t)$ | | | | | | | | |

$t$ | $\frac{7\pi}{6}$ | $\frac{5\pi}{4}$ | $\frac{4\pi}{3}$ | $\frac{3\pi}{2}$ | $\frac{5\pi}{3}$ | $\frac{7\pi}{4}$ | $\frac{11\pi}{6}$ | $2\pi$

$\sin(t)$ | $-\frac{1}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{\sqrt{3}}{2}$ | $-1$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{1}{2}$ | $0$

$\cos(t)$ | $-\frac{\sqrt{3}}{2}$ | $-\frac{\sqrt{2}}{2}$ | $-\frac{1}{2}$ | $0$ | $\frac{1}{2}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{3}}{2}$ | $1$

$\tan(t)$ | $\frac{1}{\sqrt{3}}$ | $1$ | $\frac{3}{\sqrt{3}}$ | u | $-\frac{3}{\sqrt{3}}$ | $-1$ | $-\frac{1}{\sqrt{3}}$ | $0$

$\cot(t)$ | | | | | | | |

In which quadrants is $r(t) = \cot(t)$ positive? negative?

At what $t$-values does $r(t) = \cot(t)$ have a vertical asymptote? Why?

What is the domain of the cotangent function? What is its range?

Sketch an accurate, labeled graph of $r(t) = \cot(t)$ on the axes provided below, including the special points that come from the unit circle.

On intervals where the function is defined at every point in the interval, is $r(t) = \cot(t)$ always increasing, always decreasing, or neither?

What is the period of the cotangent function?

How would you describe the relationship between the graphs of the tangent and cotangent functions?