Given the following right triangle find cos0, sine tane, sece csce, and cott, Do not approximate: Find exact answers. Show allof your work and explain steps as necessary. 10 7
Given the following right triangle find cos0, sine tane, sece csce, and cott, Do not approximate: Find exact answers. Show allof your work and explain steps as necessary. 10 7
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
Your help will be really appreciated as I am at a loss of what to do. Thank you very much for your help.
- Hypotenuse: \( 10 \)
- Opposite side: \( 7 \)
- Adjacent side: \( 6 \)
### Steps and Definitions
Given a right triangle with:
- **Hypotenuse (c)**: The longest side, opposite the right angle
- **Opposite side (a)**: The side opposite to the angle \( \theta \)
- **Adjacent side (b)**: The side next to the angle \( \theta \)
The trigonometric functions are defined as follows:
1. **Cosine ( \( \cos\theta \) )**:
\[
\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{6}{10} = \frac{3}{5}
\]
2. **Sine ( \( \sin\theta \) )**:
\[
\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{10}
\]
3. **Tangent ( \( \tan\theta \) )**:
\[
\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{6}
\]
4. **Secant ( \( \sec\theta \) )**:
\[
\sec\theta = \frac{1}{\cos\theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}
\]
5. **Cosecant ( \( \csc\theta \) )**:
\[
\csc\theta = \frac{1}{\sin\theta} = \frac{1}{\frac{7}{10}} = \frac{10}{7}
\]
6. **Cotangent ( \( \cot\theta \) )**:
\[](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9902370-7e16-4b28-876b-51641093f7a2%2F2375e47c-1238-4a97-948f-7cc0f7c6c2a6%2Fmp6wcavd.jpeg&w=3840&q=75)
Transcribed Image Text:## Problem Statement
1. **Given the following right triangle, find \( \cos\theta \), \( \sin\theta \), \( \tan\theta \), \( \sec\theta \), \( \csc\theta \), and \( \cot\theta \). Do not approximate: Find exact answers. Show all of your work and explain steps as necessary.**
- Hypotenuse: \( 10 \)
- Opposite side: \( 7 \)
- Adjacent side: \( 6 \)
### Steps and Definitions
Given a right triangle with:
- **Hypotenuse (c)**: The longest side, opposite the right angle
- **Opposite side (a)**: The side opposite to the angle \( \theta \)
- **Adjacent side (b)**: The side next to the angle \( \theta \)
The trigonometric functions are defined as follows:
1. **Cosine ( \( \cos\theta \) )**:
\[
\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{6}{10} = \frac{3}{5}
\]
2. **Sine ( \( \sin\theta \) )**:
\[
\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{10}
\]
3. **Tangent ( \( \tan\theta \) )**:
\[
\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{6}
\]
4. **Secant ( \( \sec\theta \) )**:
\[
\sec\theta = \frac{1}{\cos\theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}
\]
5. **Cosecant ( \( \csc\theta \) )**:
\[
\csc\theta = \frac{1}{\sin\theta} = \frac{1}{\frac{7}{10}} = \frac{10}{7}
\]
6. **Cotangent ( \( \cot\theta \) )**:
\[
![### Prove the Identity:
\[ \cos(x - y)\cos y - \sin(x - y)\sin y = \cos x \]
---
#### [When you are finished, delete this text and the guidelines in the table below.]
| Statement | Rule |
|-----------|------|
| Use this template to indicate your answer. You can create more rows if necessary. Indicate each step of your process in this column | Defend your process by identifying the appropriate explanation for each process step in this column |
| Remember to: | |
| - Identify the problem statement. | |
| - Correctly use appropriate identities and/or theorems. | |
| - Correctly use the algebraic process. | |
| - Identify the final statement. | |](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9902370-7e16-4b28-876b-51641093f7a2%2F2375e47c-1238-4a97-948f-7cc0f7c6c2a6%2F289yf0p.jpeg&w=3840&q=75)
Transcribed Image Text:### Prove the Identity:
\[ \cos(x - y)\cos y - \sin(x - y)\sin y = \cos x \]
---
#### [When you are finished, delete this text and the guidelines in the table below.]
| Statement | Rule |
|-----------|------|
| Use this template to indicate your answer. You can create more rows if necessary. Indicate each step of your process in this column | Defend your process by identifying the appropriate explanation for each process step in this column |
| Remember to: | |
| - Identify the problem statement. | |
| - Correctly use appropriate identities and/or theorems. | |
| - Correctly use the algebraic process. | |
| - Identify the final statement. | |
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