Chapter 4.3, Problem 55E

(a)

To determine

The value of $\sin \theta$ with the help of the trigonometric identities.

Answer to Problem 55E

The value of $\sin \theta$ is $\frac{1}{3}$ .

Explanation of Solution

Given Information:

The value of $\csc \theta =3$ and $\sec \theta =\frac{3\sqrt{2}}{4}$ .

Formula Used:

Trigonometric identity $\sin \theta =\frac{1}{\csc \theta }$

Calculation:

Substitute $3$ for $\csc \theta$ in above identity.

  $\begin{array}{c}\sin \theta =\frac{1}{\csc \theta }\\ =\frac{1}{3}\end{array}$

Therefore the value of $\csc \theta$ is $\frac{1}{3}$ .

(b)

To determine

The value of $\cos \theta$ with the help of the trigonometric identities.

Answer to Problem 55E

The value of $\cos \theta$ is $\frac{2\sqrt{2}}{3}$ .

Explanation of Solution

Given Information:

The value of $\csc \theta =3$ and $\sec \theta =\frac{3\sqrt{2}}{4}$ .

Formula Used:

Trigonometric identity $\cos \theta =\frac{1}{\sec \theta }$

Calculation:

Substitute $\frac{3\sqrt{2}}{4}$ for $\sec \theta$ in above identity.

  $\begin{array}{l}\cos \theta =\frac{1}{\sec \theta }\\ =\frac{1}{\frac{3\sqrt{2}}{4}}\\ =\frac{4}{3\sqrt{2}}\\ =\frac{2\sqrt{2}}{3}\end{array}$

Therefore the value of $\sec \theta$ is $\frac{2\sqrt{2}}{3}$ .

(c)

To determine

The value of $\tan \theta$ with the help of the trigonometric identities.

Answer to Problem 55E

The value of $\tan \theta$ is $\frac{\sqrt{2}}{4}$ .

Explanation of Solution

Given Information:

The value of $\csc \theta =3$ and $\sec \theta =\frac{3\sqrt{2}}{4}$ .

Formula Used:

Trigonometric identity

  $\begin{array}{c}\tan \theta =\frac{\sec \theta }{\csc \theta }\\ =\frac{\sin \theta }{\cos \theta }\end{array}$ .

Calculation:

Substitute $\frac{3\sqrt{2}}{4}$ for $\sec \theta$ and $3$ for $\csc \theta$ in above identity.

  $\begin{array}{c}\tan \theta =\frac{\sec \theta }{\csc \theta }\\ =\frac{\frac{3\sqrt{2}}{4}}{3}\\ =\frac{\sqrt{2}}{4}\end{array}$

Therefore the value of $\tan \theta$ is $\frac{\sqrt{2}}{4}$ .

(d)

To determine

The value of $\sec \left(90°-\theta \right)$ with the help of the trigonometric identities.

Answer to Problem 55E

The value of $\sec \left(90°-\theta \right)$ is $3$ .

Explanation of Solution

Given Information:

The value of $\csc \theta =3$ and $\sec \theta =\frac{3\sqrt{2}}{4}$ .

Formula Used:

Trigonometric identity $\sec \left(90°-\theta \right)=\csc \theta$

Calculation:

Substitute $3$ for $\csc \theta$ in above identity.

  $\begin{array}{c}\sec \left(90°-\theta \right)=\csc \theta \\ =3\end{array}$

Therefore the value of $\sec \left(90°-\theta \right)$ is $3$ .