https://file.aiquickdraw.com/mathbot/file/1728964555448wsj4b9t6.jpg

The equation given is:

\[
\sin^2 \theta - 2 \sin \theta - 3 = 0
\]

We can solve this as a quadratic equation in terms of \( \sin \theta \). Let’s define a substitution:

\[
x = \sin \theta
\]

Now the equation becomes:

\[
x^2 - 2x - 3 = 0
\]

This is a standard quadratic equation, which can be factored as:

\[
(x - 3)(x + 1) = 0
\]

Thus, the solutions for \( x \) are:

\[
x = 3 \quad \text{or} \quad x = -1
\]

Since \( \sin \theta = x \), we now substitute back:

1. \( \sin \theta = 3 \): This has no solution since the sine function's range is \([-1, 1]\), and \( 3 \) is outside this range.
2. \( \sin \theta = -1 \): This is within the valid range of the sine function. The solution occurs at \( \theta = \frac{3\pi}{2} \) in the interval \( [0, 2\pi] \).

Thus, the only solution in the interval \( [0, 2\pi] \) is:

\[
\theta = \frac{3\pi}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. How would the solution change if the equation involved \( \sin^2 \theta + 2 \sin \theta - 3 = 0 \)?
2. What is the general solution for \( \sin \theta = -1 \) over all possible intervals?
3. What are the typical ranges for sine and cosine functions in radians?
4. How can we use the unit circle to solve trigonometric equations involving sine or cosine?
5. Can a similar method be applied for equations involving \( \cos^2 \theta \)?

**Tip**: Always check the range of trigonometric functions when solving equations, as some solutions might not be valid.

Find all solutions of the equation sin^2(θ) - 2sin(θ) - 3 = 0 in the interval [0, 2π]. Write your answer in radians in terms of π.

Learn how to solve the trigonometric equation sin^2(θ) - 2sin(θ) - 3 = 0 within the interval [0, 2π] using factoring. This problem covers key concepts in trigonometry and solving quadratic equations. Suitable for high school 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