Solve the equation on the interval 0 s0< 2x. sin (20) == 1 What are the solutions to sin (20) =, in the interval 0 s0< 2x? Select the correct choice and fill in any answer boxes in your choice below. The solution set is {} (Simplify your answer. Type an exact answer, using t as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
Solve the equation on the interval 0 s0< 2x. sin (20) == 1 What are the solutions to sin (20) =, in the interval 0 s0< 2x? Select the correct choice and fill in any answer boxes in your choice below. The solution set is {} (Simplify your answer. Type an exact answer, using t as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
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°.
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![### Solving Trigonometric Equations
Let's learn how to solve the trigonometric equation given on the interval \( 0 \leq \theta < 2\pi \).
**Equation:**
\[ \sin(2\theta) = \frac{1}{2} \]
**Question:**
What are the solutions to \(\sin(2\theta) = \frac{1}{2} \) in the interval \( 0 \leq \theta < 2\pi \)?
Select the correct choice and fill in any answer boxes in your choice below.
**Solution:**
The solution set is
\[ \{ \square, \square \} \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
**Explanation:**
To solve for \(\theta\) in the equation \(\sin(2\theta) = \frac{1}{2}\), we identify the angles where the sine of an angle equals \( \frac{1}{2} \). Remember that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
\[ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \]
Thus:
\[ 2\theta = \frac{\pi}{6} \]
\[ 2\theta = \frac{5\pi}{6} \]
Solving for \(\theta\):
\[ \theta = \frac{\pi}{12} \]
\[ \theta = \frac{5\pi}{12} \]
Additionally, since the sine function has a period of \( 2\pi \), we must consider additional solutions within the interval by including \( 2\pi k \), where \( k \) is an integer:
\[ \theta = \frac{\pi}{12} + \pi k \]
\[ \theta = \frac{5\pi}{12} + \pi k \]
We find the specific solutions within the given range \( 0 \leq \theta < 2\pi \).
### Conclusion
The exact solutions, appropriately simplified, would be:
\[ \theta = \left\{\frac{\pi}{12}, \frac{5\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56b89a6c-6ca3-4f25-9a44-1409c69966fc%2F6daf62f8-0868-4ec7-9297-f5ab350bacad%2Fagfiw6m_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Trigonometric Equations
Let's learn how to solve the trigonometric equation given on the interval \( 0 \leq \theta < 2\pi \).
**Equation:**
\[ \sin(2\theta) = \frac{1}{2} \]
**Question:**
What are the solutions to \(\sin(2\theta) = \frac{1}{2} \) in the interval \( 0 \leq \theta < 2\pi \)?
Select the correct choice and fill in any answer boxes in your choice below.
**Solution:**
The solution set is
\[ \{ \square, \square \} \]
(Simplify your answer. Type an exact answer, using \(\pi\) as needed. Type your answer in radians. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)
**Explanation:**
To solve for \(\theta\) in the equation \(\sin(2\theta) = \frac{1}{2}\), we identify the angles where the sine of an angle equals \( \frac{1}{2} \). Remember that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
\[ \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2} \]
Thus:
\[ 2\theta = \frac{\pi}{6} \]
\[ 2\theta = \frac{5\pi}{6} \]
Solving for \(\theta\):
\[ \theta = \frac{\pi}{12} \]
\[ \theta = \frac{5\pi}{12} \]
Additionally, since the sine function has a period of \( 2\pi \), we must consider additional solutions within the interval by including \( 2\pi k \), where \( k \) is an integer:
\[ \theta = \frac{\pi}{12} + \pi k \]
\[ \theta = \frac{5\pi}{12} + \pi k \]
We find the specific solutions within the given range \( 0 \leq \theta < 2\pi \).
### Conclusion
The exact solutions, appropriately simplified, would be:
\[ \theta = \left\{\frac{\pi}{12}, \frac{5\
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