Chapter 5, Problem 7CR

To determine

To calculate: The values of the remaining trigonometric functions.

Answer to Problem 7CR

The required values of $\sin x$ , $\cos x$ , $\cot x$ , $\sec x$ and $\csc x$ are $\frac{4}{5}$ , $\frac{3}{5}$ , $\frac{3}{4}$ , $\frac{5}{3}$ and $\frac{5}{4}$ respectively.

Explanation of Solution

Given information:

The function: $\tan x=\frac{4}{3}$ , $\sec x<0$

Formula used:

Pythagoras Theorem:

In a right-angle triangle,

  $\begin{array}{c}{\text{hypotenuse}}^{\text{2}}={\text{perpendicular}}^2+{\text{base}}^2\\ h^2=p^2+b^2\end{array}$

Calculation:

The given expression is $\tan x=\frac{4}{3}$ .

So, $\cot x=\frac{3}{4}$

It is known that $\tan \theta =\frac{p}{b}$ .

Use the Pythagoras Theorem $h^2=p^2+b^2$ .

  $\begin{array}{c}{h}^2=4^2+3^2\\ h=\sqrt{16+9}\\ h=\sqrt{25}\\ h=5\end{array}$

Now, $\begin{array}{c}\sin x=\frac{p}{h}\\ =\frac{4}{5}\end{array}$

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

Also, $\begin{array}{c}\cos x=\frac{b}{h}\\ =\frac{3}{5}\end{array}$

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

Hence, the required values of $\sin x$ , $\cos x$ , $\cot x$ , $\sec x$ and $\csc x$ are $\frac{4}{5}$ , $\frac{3}{5}$ , $\frac{3}{4}$ , $\frac{5}{3}$ and $\frac{5}{4}$ respectively.