If 8 is in the third quadrant and sin 8 = -3, what is cos(0)? 5' Determine your answer without using a calculator. Rationalize the denominator if necessary. [? cos (0) = - Enter your answer in simplest form. Enter
If 8 is in the third quadrant and sin 8 = -3, what is cos(0)? 5' Determine your answer without using a calculator. Rationalize the denominator if necessary. [? cos (0) = - Enter your answer in simplest form. Enter
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°.
Related questions
Question
![### Trigonometric Problem:
**Problem Statement:**
If θ is in the third quadrant and sin(θ) = -3/5, what is cos(θ/2)?
**Instructions:**
Determine your answer without using a calculator. Rationalize the denominator if necessary.
**Given Information:**
- \( \sin(\theta) = -\frac{3}{5} \)
- θ is in the third quadrant
**Form to Fill:**
\[ \cos\left(\frac{\theta}{2}\right) = -\sqrt{\boxed{\phantom{?}}} \]
Enter your answer in simplest form.
**Input Field:**
There is a text box provided to enter the simplified answer followed by a blue "Enter" button to submit.
**Explanation of the Diagrams:**
- There is a mathematical expression format shown involving a square root and a box to input the content under the square root.
### Educational Explanation:
To solve this problem, recall the required identities and properties of trigonometric functions. In the third quadrant, the sine is negative and the cosine is also negative. Additionally, you will need to use half-angle identities:
For \( \cos\left(\frac{\theta}{2}\right) \):
\[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \]
Given that θ is in the third quadrant, you need to correctly establish the sign based on the quadrant properties for the half-angle.
To find \( \cos(\theta) \) when \( \sin(\theta) \) is known:
\[ \sin(\theta) = -\frac{3}{5} \]
Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \left( -\frac{3}{5} \right)^2 + \cos^2(\theta) = 1 \]
\[ \frac{9}{25} + \cos^2(\theta) = 1 \]
\[ \cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ \cos(\theta) = -\frac{4}{5} \] (since cosine is negative in the third quadrant)
Now, apply these to the half-angle identity:
\[ \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6990879d-8b80-4fc7-869f-9729211cbc42%2F51c63e80-6b9e-4967-aea0-d5f6031d380e%2F7forjtb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Problem:
**Problem Statement:**
If θ is in the third quadrant and sin(θ) = -3/5, what is cos(θ/2)?
**Instructions:**
Determine your answer without using a calculator. Rationalize the denominator if necessary.
**Given Information:**
- \( \sin(\theta) = -\frac{3}{5} \)
- θ is in the third quadrant
**Form to Fill:**
\[ \cos\left(\frac{\theta}{2}\right) = -\sqrt{\boxed{\phantom{?}}} \]
Enter your answer in simplest form.
**Input Field:**
There is a text box provided to enter the simplified answer followed by a blue "Enter" button to submit.
**Explanation of the Diagrams:**
- There is a mathematical expression format shown involving a square root and a box to input the content under the square root.
### Educational Explanation:
To solve this problem, recall the required identities and properties of trigonometric functions. In the third quadrant, the sine is negative and the cosine is also negative. Additionally, you will need to use half-angle identities:
For \( \cos\left(\frac{\theta}{2}\right) \):
\[ \cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} \]
Given that θ is in the third quadrant, you need to correctly establish the sign based on the quadrant properties for the half-angle.
To find \( \cos(\theta) \) when \( \sin(\theta) \) is known:
\[ \sin(\theta) = -\frac{3}{5} \]
Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \left( -\frac{3}{5} \right)^2 + \cos^2(\theta) = 1 \]
\[ \frac{9}{25} + \cos^2(\theta) = 1 \]
\[ \cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \]
\[ \cos(\theta) = -\frac{4}{5} \] (since cosine is negative in the third quadrant)
Now, apply these to the half-angle identity:
\[ \
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