Chapter 4.4, Problem 15E

To determine

To find: The exact value of six trigonometric function of each angle $\theta$ .

Answer to Problem 15E

The exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =\frac{7}{25},\sin \theta =\frac{24}{25},\cot \theta =\frac{7}{24},\tan \theta =\frac{24}{7},$$\sec \theta =\frac{25}{7},\csc \theta =\frac{25}{24}$ .

Explanation of Solution

Given information:

The given point is on the terminal side of an angle in standard position is $\left(7,24\right)$ .

Calculation:

Use $\theta$ be an angle in standard position with $\left(x,y\right)$ from given points $\left(7,24\right)$ .

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ \ne 0\end{array}$

The trigonometric functions of any angle.

  $\begin{array}{l}\cos \theta =\frac{x}{r}\\ \sin \theta =\frac{y}{r}\\ \cot \theta =\frac{x}{y}\\ \tan \theta =\frac{y}{x}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{r}{x}\\ \csc \theta =\frac{r}{y}\end{array}$

Calculate the value of $r$ and then calculate the value of the trigonometric functions.

  $\begin{array}{c}r=\sqrt{x^2+y^2}\\ =\sqrt{7^2+{24}^2}\\ =25\end{array}$

Substitute $7$ for $x$ , $24$ for $y$ , and $25$ for $r$ in trigonometric functions.

  $\begin{array}{l}\cos \theta =\frac{7}{25}\\ \sin \theta =\frac{24}{25}\\ \cot \theta =\frac{7}{24}\\ \tan \theta =\frac{24}{7}\end{array}$

  $\begin{array}{l}\sec \theta =\frac{25}{7}\\ \csc \theta =\frac{25}{24}\end{array}$

Therefore, the exact value of six trigonometric function of each angle $\theta$ are $\cos \theta =\frac{7}{25},\sin \theta =\frac{24}{25},\cot \theta =\frac{7}{24},\tan \theta =\frac{24}{7},$$\sec \theta =\frac{25}{7},\csc \theta =\frac{25}{24}$ .