4 and 90° <0 < 180°. Suppose that sin0 = Find the exact values of sin and tan
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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![### Trigonometric Identities and Half-Angle Formulas
#### Problem Statement:
1. Suppose that:
\[ \sin{\theta} = \frac{4}{5} \]
and
\[ 90^\circ < \theta < 180^\circ. \]
2. Find the exact values of:
\[ \sin{\left(\frac{\theta}{2}\right)} \]
and
\[ \tan{\left(\frac{\theta}{2}\right)}. \]
#### Graphical Components:
The provided image includes text and a portion of an equation editor which suggests the visual process of solving the problem using software.
#### Step-by-Step Solution:
To find the values of \(\sin{\left(\frac{\theta}{2}\right)}\) and \(\tan{\left(\frac{\theta}{2}\right)}\), we use the half-angle formulas:
\[ \sin{\left(\frac{\theta}{2}\right)} = \sqrt{\frac{1 - \cos{\theta}}{2}} \]
\[ \tan{\left(\frac{\theta}{2}\right)} = \pm \sqrt{\frac{1 - \cos{\theta}}{1 + \cos{\theta}}},\]
based on the quadrant of \(\theta/2 \).
First, we need to find \(\cos{\theta}\).
Given that:
\[ \sin{\theta} = \frac{4}{5}, \]
we can use the Pythagorean identity:
\[ \sin^2{\theta} + \cos^2{\theta} = 1. \]
\[ \left(\frac{4}{5}\right)^2 + \cos^2{\theta} = 1. \]
\[ \frac{16}{25} + \cos^2{\theta} = 1. \]
\[ \cos^2{\theta} = 1 - \frac{16}{25}. \]
\[ \cos^2{\theta} = \frac{25}{25} - \frac{16}{25}. \]
\[ \cos^2{\theta} = \frac{9}{25}. \]
\[ \cos{\theta} = \pm \frac{3}{5}. \]
For \( 90^\circ < \theta < 180^\circ \), \(\cos{\theta}\) is negative.
\[ \cos{\theta} = -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0b581ba-098c-432d-a731-9ec346ce3593%2F3a678233-6fa0-4f1f-9d77-32c8f032f967%2F5r3gbuf_processed.png&w=3840&q=75)
Transcribed Image Text:### Trigonometric Identities and Half-Angle Formulas
#### Problem Statement:
1. Suppose that:
\[ \sin{\theta} = \frac{4}{5} \]
and
\[ 90^\circ < \theta < 180^\circ. \]
2. Find the exact values of:
\[ \sin{\left(\frac{\theta}{2}\right)} \]
and
\[ \tan{\left(\frac{\theta}{2}\right)}. \]
#### Graphical Components:
The provided image includes text and a portion of an equation editor which suggests the visual process of solving the problem using software.
#### Step-by-Step Solution:
To find the values of \(\sin{\left(\frac{\theta}{2}\right)}\) and \(\tan{\left(\frac{\theta}{2}\right)}\), we use the half-angle formulas:
\[ \sin{\left(\frac{\theta}{2}\right)} = \sqrt{\frac{1 - \cos{\theta}}{2}} \]
\[ \tan{\left(\frac{\theta}{2}\right)} = \pm \sqrt{\frac{1 - \cos{\theta}}{1 + \cos{\theta}}},\]
based on the quadrant of \(\theta/2 \).
First, we need to find \(\cos{\theta}\).
Given that:
\[ \sin{\theta} = \frac{4}{5}, \]
we can use the Pythagorean identity:
\[ \sin^2{\theta} + \cos^2{\theta} = 1. \]
\[ \left(\frac{4}{5}\right)^2 + \cos^2{\theta} = 1. \]
\[ \frac{16}{25} + \cos^2{\theta} = 1. \]
\[ \cos^2{\theta} = 1 - \frac{16}{25}. \]
\[ \cos^2{\theta} = \frac{25}{25} - \frac{16}{25}. \]
\[ \cos^2{\theta} = \frac{9}{25}. \]
\[ \cos{\theta} = \pm \frac{3}{5}. \]
For \( 90^\circ < \theta < 180^\circ \), \(\cos{\theta}\) is negative.
\[ \cos{\theta} = -
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