**Mathematics Practice** Below are a few mathematical problems requiring you to find values and establish identities. --- **Problem 4:** \[ \cos 2\theta = \frac{24}{25}, \quad \frac{\pi}{2} < 2\theta < \pi \] Find \(\sin \theta\). Options: - A) \(\frac{7}{5}\) - B) \(\frac{7\sqrt{2}}{10}\) - C) \(-\frac{7\sqrt{2}}{10}\) - D) \(-\frac{7}{5}\) --- **Problem 5: Find the exact value of the expression.** \[ \sin \left( 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \right) \] Options: - A) \(-\frac{\sqrt{3}}{2}\) - B) \(\sqrt{3}\) - C) \(\frac{\sqrt{3}}{2}\) - D) \(\frac{1}{2}\) --- **Problem 6: Establish the identity.** \[ \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right)^2 = 1 - \sin x \] --- For each problem, select the correct option and verify your answer with suitable calculations or reasoning based on trigonometric identities and properties.
**Mathematics Practice** Below are a few mathematical problems requiring you to find values and establish identities. --- **Problem 4:** \[ \cos 2\theta = \frac{24}{25}, \quad \frac{\pi}{2} < 2\theta < \pi \] Find \(\sin \theta\). Options: - A) \(\frac{7}{5}\) - B) \(\frac{7\sqrt{2}}{10}\) - C) \(-\frac{7\sqrt{2}}{10}\) - D) \(-\frac{7}{5}\) --- **Problem 5: Find the exact value of the expression.** \[ \sin \left( 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \right) \] Options: - A) \(-\frac{\sqrt{3}}{2}\) - B) \(\sqrt{3}\) - C) \(\frac{\sqrt{3}}{2}\) - D) \(\frac{1}{2}\) --- **Problem 6: Establish the identity.** \[ \left( \cos \frac{x}{2} - \sin \frac{x}{2} \right)^2 = 1 - \sin x \] --- For each problem, select the correct option and verify your answer with suitable calculations or reasoning based on trigonometric identities and properties.
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
![**Mathematics Practice**
Below are a few mathematical problems requiring you to find values and establish identities.
---
**Problem 4:**
\[
\cos 2\theta = \frac{24}{25}, \quad \frac{\pi}{2} < 2\theta < \pi
\]
Find \(\sin \theta\).
Options:
- A) \(\frac{7}{5}\)
- B) \(\frac{7\sqrt{2}}{10}\)
- C) \(-\frac{7\sqrt{2}}{10}\)
- D) \(-\frac{7}{5}\)
---
**Problem 5: Find the exact value of the expression.**
\[
\sin \left( 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \right)
\]
Options:
- A) \(-\frac{\sqrt{3}}{2}\)
- B) \(\sqrt{3}\)
- C) \(\frac{\sqrt{3}}{2}\)
- D) \(\frac{1}{2}\)
---
**Problem 6: Establish the identity.**
\[
\left( \cos \frac{x}{2} - \sin \frac{x}{2} \right)^2 = 1 - \sin x
\]
---
For each problem, select the correct option and verify your answer with suitable calculations or reasoning based on trigonometric identities and properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae7608d-7f6b-4ed8-94b2-0b7eb11e0ad9%2Fb8b8cf13-c91c-42f8-b9b1-85340e81d915%2Fysy8cd_processed.png&w=3840&q=75)
Transcribed Image Text:**Mathematics Practice**
Below are a few mathematical problems requiring you to find values and establish identities.
---
**Problem 4:**
\[
\cos 2\theta = \frac{24}{25}, \quad \frac{\pi}{2} < 2\theta < \pi
\]
Find \(\sin \theta\).
Options:
- A) \(\frac{7}{5}\)
- B) \(\frac{7\sqrt{2}}{10}\)
- C) \(-\frac{7\sqrt{2}}{10}\)
- D) \(-\frac{7}{5}\)
---
**Problem 5: Find the exact value of the expression.**
\[
\sin \left( 2 \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \right)
\]
Options:
- A) \(-\frac{\sqrt{3}}{2}\)
- B) \(\sqrt{3}\)
- C) \(\frac{\sqrt{3}}{2}\)
- D) \(\frac{1}{2}\)
---
**Problem 6: Establish the identity.**
\[
\left( \cos \frac{x}{2} - \sin \frac{x}{2} \right)^2 = 1 - \sin x
\]
---
For each problem, select the correct option and verify your answer with suitable calculations or reasoning based on trigonometric identities and properties.
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