Chapter 5, Problem 11RE

To determine

To evaluate: the remaining trigonometric function of the angle..

Answer to Problem 11RE

  $\tan x=\frac{4}{3},\cot x=\frac{3}{4},\sec x=\frac{5}{3},\csc x=\frac{5}{4}$ .

Explanation of Solution

Given information:

Given trigonometric function is

  $\sin x=\frac{4}{5},\cos x=\frac{3}{5}$ .

Calculation:

Find all six trigonometric functions by applying quotient identity;

  $\tan u=\frac{\sin u}{\cos u}$

Therefore;

  $\tan x=\frac{\sin x}{\cos x}$

Substitute $\sin x=\frac{4}{5},\cos x=\frac{3}{5}$ in the above function;

  $\begin{array}{c}\tan x=\frac{\frac{4}{5}}{\frac{3}{5}}\\ =\frac{4}{\cancel{5}}\frac{\cancel{5}}{3}\\ =\frac{4}{3}\end{array}$

Apply reciprocal identity;

  $\begin{array}{l}\cot u=\frac{1}{\tan u}\\ \cot x=\frac{1}{\tan x}\end{array}$

Substitute $\tan x=\frac{4}{3}$ in the above function;

  $\begin{array}{c}\cot x=\frac{1}{\frac{4}{3}}\\ =\frac{3}{4}\end{array}$

Apply reciprocal identity $\sec u=\frac{1}{\cos u}$

  $\sec x=\frac{1}{\cos x}$

Substitute $\cos x=\frac{3}{5}$ in the above expression;

  $\begin{array}{c}\sec x=\frac{1}{\frac{3}{5}}\\ =\frac{5}{3}\end{array}$

Apply reciprocal identity $\csc u=\frac{1}{\sin u}$

  $\csc x=\frac{1}{\sin x}$

Substitute $\sin x=\frac{4}{5}$ in the above function;

  $\begin{array}{c}\csc x=\frac{1}{\sin x}\\ =\frac{1}{\frac{4}{5}}\\ =\frac{5}{4}\end{array}$

Therefore remaining trigonometric functions are $\tan x=\frac{4}{3},\cot x=\frac{3}{4},\sec x=\frac{5}{3},\csc x=\frac{5}{4}$ .