Question Given that sin(0) = . and is in Quadrant II, what is cos(20)? Provide your answer below: cos (20)= Content attribution Q Search hp FEEDE
Question Given that sin(0) = . and is in Quadrant II, what is cos(20)? Provide your answer below: cos (20)= Content attribution Q Search hp FEEDE
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
![### Trigonometry Question:
#### Problem Statement:
Given that \( \sin(\theta) = \frac{4}{5} \) and \( \theta \) is in Quadrant II, what is \( \cos(2\theta) \)?
#### Solution:
**Step 1: Determine \(\cos(\theta)\) in Quadrant II:**
In Quadrant II, \(\cos(\theta)\) is negative. Using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\):
\[
\sin(\theta) = \frac{4}{5}
\]
\[
\sin^2(\theta) = \left(\frac{4}{5}\right)^2 = \frac{16}{25}
\]
\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{16}{25} = \frac{9}{25}
\]
Since \(\theta\) is in Quadrant II, \(\cos(\theta)\) is negative:
\[
\cos(\theta) = -\sqrt{\frac{9}{25}} = -\frac{3}{5}
\]
**Step 2: Calculate \(\cos(2\theta)\) using the double-angle formula:**
The double-angle formula for cosine is:
\[
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\]
Substitute the `values`:
\[
\cos(2\theta) = \left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = \frac{9 - 16}{25} = -\frac{7}{25}
\]
**Step 3: Answer the Question:**
Therefore, the value of \( \cos(2\theta) \) is:
\[
\cos(2\theta) = -\frac{7}{25}
\]
#### Answer Submission:
cos(2θ) = **-7/25**
Feel free to use the feedback option if you have any questions or need further assistance!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ede6f4d-bbcd-4ef2-8e84-4e31e223401b%2Fc63ab11d-0079-4f2a-a2b4-421e768c4758%2Flel8mdo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometry Question:
#### Problem Statement:
Given that \( \sin(\theta) = \frac{4}{5} \) and \( \theta \) is in Quadrant II, what is \( \cos(2\theta) \)?
#### Solution:
**Step 1: Determine \(\cos(\theta)\) in Quadrant II:**
In Quadrant II, \(\cos(\theta)\) is negative. Using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\):
\[
\sin(\theta) = \frac{4}{5}
\]
\[
\sin^2(\theta) = \left(\frac{4}{5}\right)^2 = \frac{16}{25}
\]
\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{16}{25} = \frac{9}{25}
\]
Since \(\theta\) is in Quadrant II, \(\cos(\theta)\) is negative:
\[
\cos(\theta) = -\sqrt{\frac{9}{25}} = -\frac{3}{5}
\]
**Step 2: Calculate \(\cos(2\theta)\) using the double-angle formula:**
The double-angle formula for cosine is:
\[
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)
\]
Substitute the `values`:
\[
\cos(2\theta) = \left(-\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = \frac{9 - 16}{25} = -\frac{7}{25}
\]
**Step 3: Answer the Question:**
Therefore, the value of \( \cos(2\theta) \) is:
\[
\cos(2\theta) = -\frac{7}{25}
\]
#### Answer Submission:
cos(2θ) = **-7/25**
Feel free to use the feedback option if you have any questions or need further assistance!
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