### Problem DescriptionYou are given the following trigonometric equation:\[\sin a = \frac{11}{61}, \quad 0 < a < \frac{\pi}{2}\]### TaskCalculate the following half-angle trigonometric functions:#### Part 1 of 3(a) Compute $\sin \frac{a}{2}$.\[ \sin \frac{a}{2} = \square \]#### Part 2 of 3(b) Compute $\cos \frac{a}{2}$.\[ \cos \frac{a}{2} = \square \]#### Part 3 of 3(c) Compute $\tan \frac{a}{2}$.\[ \tan \frac{a}{2} = \square \]### InstructionsUse the half-angle formulas in trigonometry to solve for each part. The formulas are:- $\sin \frac{a}{2} = \pm \sqrt{\frac{1 - \cos a}{2}}$- $\cos \frac{a}{2} = \pm \sqrt{\frac{1 + \cos a}{2}}$- $\tan \frac{a}{2} = \frac{1 - \cos a}{\sin a} = \frac{\sin a}{1 + \cos a}$Ensure to verify the sign based on the given range for $a$.

Name: ### Problem DescriptionYou are given the following trigonometric equa
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Description: ### Problem DescriptionYou are given the following trigonometric equation:\[\sin a = \frac{11}{61}, \quad 0 < a < \frac{\pi}{2}\]### TaskCalculate the following half-angle trigonometric functions:#### Part 1 of 3(a) Compute $\sin \frac{a}{2}$.\[ \

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Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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### Problem Description

You are given the following trigonometric equation:

\[
\sin a = \frac{11}{61}, \quad 0 < a < \frac{\pi}{2}
\]

### Task

Calculate the following half-angle trigonometric functions:

#### Part 1 of 3

(a) Compute $\sin \frac{a}{2}$.

\[ \sin \frac{a}{2} = \square \]

#### Part 2 of 3

(b) Compute $\cos \frac{a}{2}$.

\[ \cos \frac{a}{2} = \square \]

#### Part 3 of 3

(c) Compute $\tan \frac{a}{2}$.

\[ \tan \frac{a}{2} = \square \]

### Instructions

Use the half-angle formulas in trigonometry to solve for each part. The formulas are:

- $\sin \frac{a}{2} = \pm \sqrt{\frac{1 - \cos a}{2}}$
- $\cos \frac{a}{2} = \pm \sqrt{\frac{1 + \cos a}{2}}$
- $\tan \frac{a}{2} = \frac{1 - \cos a}{\sin a} = \frac{\sin a}{1 + \cos a}$

Ensure to verify the sign based on the given range for $a$.

Expert Solution

Step 1

Calculate the valuer for angle $α$ using the inverse of the sine function.

$\begin{array}{l} \sin α = \frac{11}{61} \\ α = \sin^{- 1} (\frac{11}{61}) \\ α = \sin^{- 1} (0.18) \\ α = 10.389 ° \end{array}$

Then, determine the values for $\begin{array}{l} \sin \frac{α}{2} a n d \cos \frac{α}{2} \end{array}$ .

(a)

$\begin{matrix} \sin \frac{α}{2} = \sin (\frac{10.389 °}{2}) \\ = \sin (5.194 °) \\ = 0.0905 \\ S o, \sin \frac{α}{2} = 0.0905 \end{matrix}$

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