Chapter 5.1, Problem 1E

(a)

To determine

To match the function $\sin u$ with an equivalent expression.

Answer to Problem 1E

  $\left(iii\right)\frac{1}{\csc u}$

Explanation of Solution

Given:

The expressions,

  $\begin{array}{l}\left(i\right)\frac{1}{\sec u}\\ \left(ii\right)\frac{1}{\cot u}\\ \left(iii\right)\frac{1}{\csc u}\end{array}$

Concept Used:

Reciprocal identities of trigonometric functions:

  $\begin{array}{l}\sin \theta =\frac{1}{\csc \theta },\csc \theta =\frac{1}{\sin \theta },\\ \cos \theta =\frac{1}{\sec \theta },\sec \theta =\frac{1}{\cos \theta },\\ \tan \theta =\frac{1}{\cot \theta },\cot \theta =\frac{1}{\tan \theta }\end{array}$

Calculation:

In order tofind the equivalent expression for the function $\sin u$ , use the reciprocal identities of the trigonometric function.

Observe that the csc is reciprocal of the sine function, so the equivalent expression for $\sin u$ is given by $\frac{1}{\csc u}$ .

Thus, the equivalent expression to $\sin u$ is $\frac{1}{\csc u}$ .

(b)

To determine

To match the function $\cos u$ with an equivalent expression.

Answer to Problem 1E

  $\left(i\right)\frac{1}{\sec u}$

Explanation of Solution

Given:

The expressions,

  $\begin{array}{l}\left(i\right)\frac{1}{\sec u}\\ \left(ii\right)\frac{1}{\cot u}\\ \left(iii\right)\frac{1}{\csc u}\end{array}$

Concept Used:

Reciprocal identities of trigonometric functions:

  $\begin{array}{l}\sin \theta =\frac{1}{\csc \theta },\csc \theta =\frac{1}{\sin \theta },\\ \cos \theta =\frac{1}{\sec \theta },\sec \theta =\frac{1}{\cos \theta },\\ \tan \theta =\frac{1}{\cot \theta },\cot \theta =\frac{1}{\tan \theta }\end{array}$

Calculation:

In order to find the equivalent expression for the function $\cos u$ , use the reciprocal identities of the trigonometric function.

Observe that the sec is reciprocal of the cosine function, so the equivalent expression for $\cos u$ is given by $\frac{1}{\sec u}$ .

Thus, the equivalent expression to $\cos u$ is $\frac{1}{\sec u}$ .

(c)

To determine

To match the function $\tan u$ with an equivalent expression.

Answer to Problem 1E

  $\left(ii\right)\frac{1}{\cot u}$

Explanation of Solution

Given:

The expressions,

  $\begin{array}{l}\left(i\right)\frac{1}{\sec u}\\ \left(ii\right)\frac{1}{\cot u}\\ \left(iii\right)\frac{1}{\csc u}\end{array}$

Concept Used:

Reciprocal identities of trigonometric functions:

  $\begin{array}{l}\sin \theta =\frac{1}{\csc \theta },\csc \theta =\frac{1}{\sin \theta },\\ \cos \theta =\frac{1}{\sec \theta },\sec \theta =\frac{1}{\cos \theta },\\ \tan \theta =\frac{1}{\cot \theta },\cot \theta =\frac{1}{\tan \theta }\end{array}$

Calculation:

In order to find the equivalent expression for the function $\tan u$ , use the reciprocal identities of the trigonometric function.

Observe that the cotangent is reciprocal of the tangent function, so the equivalent expression for $\tan u$ is given by $\frac{1}{\cot u}$ .

Thus, the equivalent expression to $\tan u$ is $\frac{1}{\cot u}$ .