Chapter 7, Problem 1RCC

(a)

To determine

To state: The reciprocal identities.

Answer to Problem 1RCC

The reciprocal identities of each trigonometry function are explained below.

Explanation of Solution

Given information:

The identity is reciprocal.

Calculation:

The reciprocal identities of each trigonometry function are as follows.

The reciprocal of sine is cosecant: $\frac{1}{\sin \theta }=\csc \theta$ .

The reciprocal of cosine is secant: $\frac{1}{\cos \theta }=\sec \theta$ .

The reciprocal of tangent is cotangent: $\frac{1}{\tan \theta }=\cot \theta$ .

The reciprocal of cotangent is tangent: $\frac{1}{\cot \theta }=\tan \theta$ .

The reciprocal of secant is cosine: $\frac{1}{\sec \theta }=\cos \theta$ .

The reciprocal of cosecant is sine: $\frac{1}{\csc \theta }=\sin \theta$ .

(b)

To determine

To state: The Pythagorean identities.

Answer to Problem 1RCC

The Pythagorean identities of each trigonometry function are explained below.

Explanation of Solution

Given information:

The identity is Pythagorean.

Calculation:

The Pythagorean identities of each trigonometry function are as follows.

The sum of square of sine and cosine is equal to 1.

  ${\sin }^{2}x+{\cos }^{2}x=1$

The sum of square of tangent and 1 is equal to ${\sec }^{2}x$ .

  ${\tan }^2+1={\sec }^{2}x$

The sum of square of cotangent and 1 is equal to ${\csc }^{2}x$ .

  $1+{\cot }^{2}x={\csc }^{2}x$

(c)

To determine

To state: The Even-Odd identities.

Answer to Problem 1RCC

The Even-Odd identities of each trigonometry function are explained below.

Explanation of Solution

Given information:

The identity is Even-Odd.

Calculation:

The Even-Odd identities of each trigonometry function are as follows.

  $\begin{array}{c}\sin \left(-x\right)=-\sin x\\ \cos \left(-x\right)=\cos x\\ \tan \left(-x\right)=-\tan x\end{array}$

(d)

To determine

To state: The co function identities.

Answer to Problem 1RCC

The co-function identities of each trigonometry function are explained below.

Explanation of Solution

Given information:

The identity is co-function.

Calculation:

The co-function identities of each trigonometry function are as follows.

  $\begin{array}{c}\sin \left(\frac{\pi }{2}-u\right)=\cos u\\ \cos \left(\frac{\pi }{2}-u\right)=\sin u\\ \tan \left(\frac{\pi }{2}-u\right)=\cot u\\ \cot \left(\frac{\pi }{2}-u\right)=\tan \end{array}$

and,

  $\begin{array}{c}\sec \left(\frac{\pi }{2}-u\right)=\csc u\\ \csc \left(\frac{\pi }{2}-u\right)=\sec u\end{array}$