(5) Suppose that 2 is an angle in quadrant 2 and that cos B Compute the exact value of cos 25 Cos ||
(5) Suppose that 2 is an angle in quadrant 2 and that cos B Compute the exact value of cos 25 Cos ||
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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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![**Problem Statement:**
Suppose that \(\frac{\beta}{2}\) is an angle in quadrant 2 and that \(\cos \beta = -\frac{7}{25}\). Compute the exact value of \(\cos \left( \frac{\beta}{2} \right)\).
---
**Solution:**
\[
\cos \left( \frac{\beta}{2} \right) = \boxed{\text{ }}
\]
In this problem, we're given that angle \(\beta\) has a cosine of \(-\frac{7}{25}\) and this angle is in the second quadrant. We need to determine the cosine of \(\frac{\beta}{2}\).
### Steps:
1. **Identifying the quadrant:**
- \(\frac{\beta}{2}\) is given to be in the second quadrant.
2. **Cosine double angle formula:**
- The cosine double angle formula can be written as:
\[
\cos 2\theta = 2\cos^2\theta - 1
\]
- Since we are working with dividing angles by 2 instead of doubling, we use the half-angle formula:
\[
\cos\left(\frac{\beta}{2}\right) = \pm \sqrt{\frac{1 + \cos \beta}{2}}
\]
- As \(\frac{\beta}{2}\) is in the second quadrant, cosine is negative in this quadrant.
3. **Using the given cosine value:**
- Given that \(\cos \beta = -\frac{7}{25}\), substitute this into the half-angle formula.
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{1 + \left(-\frac{7}{25}\right)}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{1 - \frac{7}{25}}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{\frac{25}{25} - \frac{7}{25}}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{\frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbedff2a8-4b07-409d-bb44-569d0d3dba8b%2Fdf9c317c-1915-4e99-8ddc-8215b201f94a%2Ff2kyl_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Suppose that \(\frac{\beta}{2}\) is an angle in quadrant 2 and that \(\cos \beta = -\frac{7}{25}\). Compute the exact value of \(\cos \left( \frac{\beta}{2} \right)\).
---
**Solution:**
\[
\cos \left( \frac{\beta}{2} \right) = \boxed{\text{ }}
\]
In this problem, we're given that angle \(\beta\) has a cosine of \(-\frac{7}{25}\) and this angle is in the second quadrant. We need to determine the cosine of \(\frac{\beta}{2}\).
### Steps:
1. **Identifying the quadrant:**
- \(\frac{\beta}{2}\) is given to be in the second quadrant.
2. **Cosine double angle formula:**
- The cosine double angle formula can be written as:
\[
\cos 2\theta = 2\cos^2\theta - 1
\]
- Since we are working with dividing angles by 2 instead of doubling, we use the half-angle formula:
\[
\cos\left(\frac{\beta}{2}\right) = \pm \sqrt{\frac{1 + \cos \beta}{2}}
\]
- As \(\frac{\beta}{2}\) is in the second quadrant, cosine is negative in this quadrant.
3. **Using the given cosine value:**
- Given that \(\cos \beta = -\frac{7}{25}\), substitute this into the half-angle formula.
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{1 + \left(-\frac{7}{25}\right)}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{1 - \frac{7}{25}}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{\frac{25}{25} - \frac{7}{25}}{2}}
\]
\[
\cos\left(\frac{\beta}{2}\right) = - \sqrt{\frac{\frac
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