Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Please help it’s number 11
![### Educational Content: Trigonometry Problems
---
**Question 11: Sketch one cycle of \( y = \sin x \) between 0 and \( 2\pi \). Label your axes.**
**Answer**:
To sketch one cycle of the sine function \( y = \sin x \) between 0 and \( 2\pi \):
1. **Draw the X-Axis**: Label the x-axis from 0 to \( 2\pi \) at equal intervals (\( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \)).
2. **Draw the Y-Axis**: Label the y-axis with values from -1 to 1.
3. **Plot Key Points**:
- \( x = 0 \), \( \sin(0) = 0 \)
- \( x = \frac{\pi}{2} \), \( \sin(\frac{\pi}{2}) = 1 \)
- \( x = \pi \), \( \sin(\pi) = 0 \)
- \( x = \frac{3\pi}{2} \), \( \sin(\frac{3\pi}{2}) = -1 \)
- \( x = 2\pi \), \( \sin(2\pi) = 0 \)
4. **Draw the Curve**: Connect these points with a smooth, continuous curve to form the sine wave.
This results in a single cycle of the sine function.
**Diagram Explanation**: There is no provided diagram of the sine curve in the image.
---
**Question 12: Verify the identity: \( \sin^2(x) (1 + \cot^2 x) = 1 \)**
**Answer**:
To verify the identity, let's simplify the left-hand side:
1. **Rewrite \( \cot^2 x \)**:
\[
\cot^2 x = \frac{\cos^2 x}{\sin^2 x}
\]
2. **Substitute \( \cot^2 x \) in the equation**:
\[
\sin^2 x \left(1 + \frac{\cos^2 x}{\sin^2 x}\right)
\]
3. **Simplify the expression inside the parentheses**:
\[
\sin](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3e2759e-7875-4206-9c94-eb84e9608fb1%2Fdf67c242-feca-411d-930f-16e3d40c5ce2%2Fatufa4c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content: Trigonometry Problems
---
**Question 11: Sketch one cycle of \( y = \sin x \) between 0 and \( 2\pi \). Label your axes.**
**Answer**:
To sketch one cycle of the sine function \( y = \sin x \) between 0 and \( 2\pi \):
1. **Draw the X-Axis**: Label the x-axis from 0 to \( 2\pi \) at equal intervals (\( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \)).
2. **Draw the Y-Axis**: Label the y-axis with values from -1 to 1.
3. **Plot Key Points**:
- \( x = 0 \), \( \sin(0) = 0 \)
- \( x = \frac{\pi}{2} \), \( \sin(\frac{\pi}{2}) = 1 \)
- \( x = \pi \), \( \sin(\pi) = 0 \)
- \( x = \frac{3\pi}{2} \), \( \sin(\frac{3\pi}{2}) = -1 \)
- \( x = 2\pi \), \( \sin(2\pi) = 0 \)
4. **Draw the Curve**: Connect these points with a smooth, continuous curve to form the sine wave.
This results in a single cycle of the sine function.
**Diagram Explanation**: There is no provided diagram of the sine curve in the image.
---
**Question 12: Verify the identity: \( \sin^2(x) (1 + \cot^2 x) = 1 \)**
**Answer**:
To verify the identity, let's simplify the left-hand side:
1. **Rewrite \( \cot^2 x \)**:
\[
\cot^2 x = \frac{\cos^2 x}{\sin^2 x}
\]
2. **Substitute \( \cot^2 x \) in the equation**:
\[
\sin^2 x \left(1 + \frac{\cos^2 x}{\sin^2 x}\right)
\]
3. **Simplify the expression inside the parentheses**:
\[
\sin
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