https://file.aiquickdraw.com/mathbot/file/1726797955573yxf9ugu4.jpg

The equation shown in the image is:

\[
\sin \theta = \frac{1}{2}
\]

This trigonometric equation implies that the sine of angle \(\theta\) is equal to \(\frac{1}{2}\). To solve for \(\theta\), we need to determine the angle(s) for which this equation holds true. In degrees, the standard solutions for \(\sin \theta = \frac{1}{2}\) are:

\[
\theta = 30^\circ \quad \text{or} \quad \theta = 150^\circ
\]

In radians, these values are:

\[
\theta = \frac{\pi}{6} \quad \text{or} \quad \theta = \frac{5\pi}{6}
\]

These two angles are within one full revolution (from \(0^\circ\) to \(360^\circ\) or from \(0\) to \(2\pi\) radians), but since sine is a periodic function, we can express the general solutions as:

\[
\theta = 30^\circ + 360^\circ n \quad \text{or} \quad \theta = 150^\circ + 360^\circ n
\]
where \(n\) is any integer.

In radians, this becomes:

\[
\theta = \frac{\pi}{6} + 2\pi n \quad \text{or} \quad \theta = \frac{5\pi}{6} + 2\pi n
\]
where \(n\) is any integer.

Would you like more details on the process, or do you have any questions? Here are five follow-up questions to consider:

1. What is the significance of the unit circle in finding trigonometric solutions?
2. How does sine's periodicity affect the general solution?
3. Can you determine \(\theta\) in different quadrants for other sine values?
4. What are the corresponding cosine values for these angles?
5. How do we handle angles outside the \(0^\circ\) to \(360^\circ\) range?

**Tip:** Sine functions repeat every \(360^\circ\) (or \(2\pi\) radians), which is key when solving equations involving sine.

Learn how to solve the trigonometric equation sin(θ) = 1/2 with a detailed explanation of solutions in degrees and radians. Understand the periodicity of the sine function and the general solution for trigonometric equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation