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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Question
![**Problem Statement**
Solve the equation \( \tan x - 2 \sin x = 0 \) for \( 0 \leq x \leq 2\pi \).
**Solution Approach**
To solve this trigonometric equation, we need to find the values of \( x \) that satisfy the equation within the given interval \([0, 2\pi]\).
**Steps to Solve:**
1. **Rewrite the Equation:**
\[
\tan x - 2 \sin x = 0 \implies \tan x = 2 \sin x
\]
2. **Express \(\tan x\)** in terms of \(\sin x\) and \(\cos x\):**
\[
\frac{\sin x}{\cos x} = 2 \sin x
\]
3. **Solve for Trigonometric Functions:**
We can assume \(\sin x \neq 0\), otherwise the equation would not hold as \( \tan x \) would also be zero.
\[
\frac{1}{\cos x} = 2 \quad \rightarrow \quad \cos x = \frac{1}{2}
\]
4. **Find the Angles:**
The values of \( \cos x = \frac{1}{2} \) within the interval \([0, 2\pi]\) are:
\[
x = \frac{\pi}{3}, \quad \frac{5\pi}{3}
\]
**Verification:**
Verify these solutions by substituting back into the original equation to ensure they satisfy \( \tan x = 2 \sin x \).
These steps outline the method to find and verify the solutions for the given trigonometric equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61daffe8-7e66-4f26-a5d3-c5825d31ffdf%2F4055684e-cd30-479d-bc6b-e9efc052ecf5%2F4ekobko_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Solve the equation \( \tan x - 2 \sin x = 0 \) for \( 0 \leq x \leq 2\pi \).
**Solution Approach**
To solve this trigonometric equation, we need to find the values of \( x \) that satisfy the equation within the given interval \([0, 2\pi]\).
**Steps to Solve:**
1. **Rewrite the Equation:**
\[
\tan x - 2 \sin x = 0 \implies \tan x = 2 \sin x
\]
2. **Express \(\tan x\)** in terms of \(\sin x\) and \(\cos x\):**
\[
\frac{\sin x}{\cos x} = 2 \sin x
\]
3. **Solve for Trigonometric Functions:**
We can assume \(\sin x \neq 0\), otherwise the equation would not hold as \( \tan x \) would also be zero.
\[
\frac{1}{\cos x} = 2 \quad \rightarrow \quad \cos x = \frac{1}{2}
\]
4. **Find the Angles:**
The values of \( \cos x = \frac{1}{2} \) within the interval \([0, 2\pi]\) are:
\[
x = \frac{\pi}{3}, \quad \frac{5\pi}{3}
\]
**Verification:**
Verify these solutions by substituting back into the original equation to ensure they satisfy \( \tan x = 2 \sin x \).
These steps outline the method to find and verify the solutions for the given trigonometric equation.
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