Chapter 5, Problem 135RE

To determine

The reciprocal identities, quotient identities and Pythagorean identities.

Explanation of Solution

Know that reciprocals of sine, cosine and tangent are cosecant, secant and cotangent respectively.

Write the reciprocal identities as:

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

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

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

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

  $\sec \theta =\frac{1}{\sin \theta }$

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

Also, know that tangent is the ratio of the identities sine and cosine.

So, write the quotient identities as:

  $\tan \theta =\frac{\sin \theta }{\cos \theta }$

  $\cot \theta =\frac{\cos \theta }{\sin \theta }$

Write the Pythagorean identities as:

  ${\sin }^2\theta +{\cos }^2\theta =1$

  $1+{\tan }^2\theta ={\sec }^2\theta$

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