Solve the equation for x if 0 < x < 2. Use a calculator to approximate all answers to the nearest hundr SOLUTION.) 4-4 tan(x + 3) = -12 X = X
Solve the equation for x if 0 < x < 2. Use a calculator to approximate all answers to the nearest hundr SOLUTION.) 4-4 tan(x + 3) = -12 X = X
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 59E
Related questions
Question
Section 6.1 question 22
![### Solving Trigonometric Equations
#### Problem Statement:
Solve the equation for \(x\) if \(0 \le x < 2\pi\). Use a calculator to approximate all answers to the nearest hundredth.
#### Given Equation:
\[4 - 4 \tan(x + 3) = -12\]
#### Solution:
To solve for \( x \), follow these steps:
1. **Isolate the trigonometric function:**
\[
4 - 4 \tan(x + 3) = -12
\]
Subtract 4 from both sides of the equation:
\[
-4 \tan(x + 3) = -12 - 4
\]
Simplify the right side:
\[
-4 \tan(x + 3) = -16
\]
Divide both sides by -4:
\[
\tan(x + 3) = 4
\]
2. **Use the arctangent function to find the angle:**
\[
x + 3 = \arctan(4)
\]
Using a calculator to find the arctangent of 4:
\[
x + 3 \approx 1.32 \, \text{radians}
\]
3. **Solve for \( x \):**
\[
x = 1.32 - 3
\]
Simplify:
\[
x \approx -1.68 \, \text{radians}
\]
Since the given domain is \(0 \le x < 2\pi\), we need to find a corresponding solution within this range. The value -1.68 radians can be converted to a positive equivalent within the specified range by adding \(2\pi\) (approximately 6.28):
\[
x \approx -1.68 + 2\pi
\]
\[
x \approx -1.68 + 6.28
\]
\[
x \approx 4.60 \, \text{radians}
\]
Hence, the solution to the equation is:
\[
x \approx 4.60 \, \text{radians}
\]
#### Note:
- The equation \(4 - 4 \tan(x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29eaf221-c72e-4b9c-a681-25e14c4a1cc7%2F5c506aaf-5435-478f-a9dd-98a5caec8a8f%2F39ze9xw8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Trigonometric Equations
#### Problem Statement:
Solve the equation for \(x\) if \(0 \le x < 2\pi\). Use a calculator to approximate all answers to the nearest hundredth.
#### Given Equation:
\[4 - 4 \tan(x + 3) = -12\]
#### Solution:
To solve for \( x \), follow these steps:
1. **Isolate the trigonometric function:**
\[
4 - 4 \tan(x + 3) = -12
\]
Subtract 4 from both sides of the equation:
\[
-4 \tan(x + 3) = -12 - 4
\]
Simplify the right side:
\[
-4 \tan(x + 3) = -16
\]
Divide both sides by -4:
\[
\tan(x + 3) = 4
\]
2. **Use the arctangent function to find the angle:**
\[
x + 3 = \arctan(4)
\]
Using a calculator to find the arctangent of 4:
\[
x + 3 \approx 1.32 \, \text{radians}
\]
3. **Solve for \( x \):**
\[
x = 1.32 - 3
\]
Simplify:
\[
x \approx -1.68 \, \text{radians}
\]
Since the given domain is \(0 \le x < 2\pi\), we need to find a corresponding solution within this range. The value -1.68 radians can be converted to a positive equivalent within the specified range by adding \(2\pi\) (approximately 6.28):
\[
x \approx -1.68 + 2\pi
\]
\[
x \approx -1.68 + 6.28
\]
\[
x \approx 4.60 \, \text{radians}
\]
Hence, the solution to the equation is:
\[
x \approx 4.60 \, \text{radians}
\]
#### Note:
- The equation \(4 - 4 \tan(x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning