Chapter 4, Problem 40RE

A.

To determine

Exact value $\sin \theta$ for the given data.

Answer to Problem 40RE

The value of $\sin \theta =\frac{1}{5}$

Explanation of Solution

Given:

The given value is:

  $\csc \theta =5$

As $\csc \theta =5$

So, using reciprocal identity

  $\begin{array}{l}\sin \theta =\frac{1}{\csc \theta }\\ =\frac{1}{5}\\ \Rightarrow \sin \theta =\frac{1}{5}\end{array}$

B.

To determine

Exact value $\cot \theta$ for the given data.

Answer to Problem 40RE

The value of $\cot \theta =2\sqrt{6}$

Explanation of Solution

Given:

The given value is:

  $\csc \theta =5$

Now, using identity

  ${\csc }^2\theta -{\cot }^2\theta =1$

So,

  $\begin{array}{l}{\cot }^2\theta ={\csc }^2\theta -1\\ =5^2-1\\ =24\\ \therefore \cot \theta =\sqrt{24}=2\sqrt{6}\end{array}$

C.

To determine

Exact value $\tan \theta$ for the given data.

Answer to Problem 40RE

The value of $\tan \theta =\frac{1}{2\sqrt{6}}$

Explanation of Solution

Given:

The given value is:

  $\csc \theta =5$

Using reciprocal identity

  $\tan \theta =\frac{1}{\cot \theta }$

So,

  $\tan \theta =\frac{1}{2\sqrt{6}}$

D.

To determine

Exact value $\sec \left({90}^o-\theta \right)$ for the given data.

Answer to Problem 40RE

The value of $\sec \left({90}^o-\theta \right)=5$

Explanation of Solution

Given:

The given value is:

  $\csc \theta =5$

Using the identity

  $\sec \left({90}^o-\theta \right)=\csc \theta$

As, $\csc \theta =5$

So,

  $\sec \left({90}^o-\theta \right)=5$