Chapter 5.1, Problem 12E

To determine

To calculate: The values of all six trigonometric functions.

Answer to Problem 12E

The required values of $\sin x$ , $\cos x$ , $\tan x$ , $\sec x$ and $\csc x$ are $\frac{4}{\sqrt{65}}$ , $\frac{7}{\sqrt{65}}$ , $\frac{1}{\sqrt{24}}$ , $\frac{\sqrt{65}}{7}$ and $\frac{\sqrt{65}}{4}$ respectively.

Explanation of Solution

Given information:

The functions: $\cot x=\frac{7}{4}$ , $\sin x<0$

Formula used:

Trigonometric identities:

  $\sin x=\frac{1}{\csc x}$

  $\sec x=\frac{1}{\cos x}$

  $\tan x=\frac{\sin x}{\cos x}$

  $\cot x=\frac{1}{\tan x}$

  ${\sin }^{2}x+{\cos }^{2}x=1$

  ${\sec }^{2}x=1+{\tan }^{2}x$

Calculation:

The given function is $\cot x=\frac{7}{4}$ .

So, $\begin{array}{c}\tan x=\frac{1}{\cot x}\\ =\frac{1}{\frac{7}{4}}\\ =\frac{4}{7}\end{array}$

Use the formula ${\sec }^{2}x=1+{\tan }^{2}x$ to find the value of $\sec x$ .

  $\begin{array}{c}{\sec }^{2}x=1+{\left(\frac{4}{7}\right)}^2\\ \sec x=\sqrt{1+\frac{16}{49}}\\ =\frac{\sqrt{65}}{7}\end{array}$

So, $\begin{array}{c}\cos x=\frac{1}{\sec x}\\ =\frac{1}{\frac{\sqrt{65}}{7}}\\ =\frac{7}{\sqrt{65}}\end{array}$

Use the formula ${\sin }^{2}x+{\cos }^{2}x=1$ to find the value of $\sin \varphi$ .

  $\begin{array}{c}{\sin }^{2}x+{\cos }^{2}x=1\\ {\sin }^{2}x=1-{\cos }^{2}x\\ \sin x=\sqrt{1-{\left(\frac{7}{\sqrt{65}}\right)}^2}\\ =\frac{4}{\sqrt{65}}\end{array}$

Thus, $\begin{array}{c}\csc x=\frac{1}{\sin x}\\ =\frac{1}{\frac{4}{\sqrt{65}}}\\ =\frac{\sqrt{65}}{4}\end{array}$

Hence, the required values of $\sin x$ , $\cos x$ , $\tan x$ , $\sec x$ and $\csc x$ are $\frac{4}{\sqrt{65}}$ , $\frac{7}{\sqrt{65}}$ , $\frac{1}{\sqrt{24}}$ , $\frac{\sqrt{65}}{7}$ and $\frac{\sqrt{65}}{4}$ respectively.