https://r2.erweima.ai/mathbot/image/1723781687666td3crfs6.pdf

The document you provided contains the following trigonometric problem:

\[ \text{Solve for } \theta: \quad \cos(2\theta) = \sin\left(\frac{\pi}{2} - \theta\right) \]

To solve this equation, let's proceed with the following steps:

1. **Simplify the equation:**

   \[
   \cos(2\theta) = \sin\left(\frac{\pi}{2} - \theta\right)
   \]

   We know that:
   
   \[
   \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta)
   \]
   
   Therefore, the equation becomes:
   
   \[
   \cos(2\theta) = \cos(\theta)
   \]

2. **Use the double-angle identity for cosine:**

   The identity for \(\cos(2\theta)\) is:
   
   \[
   \cos(2\theta) = 2\cos^2(\theta) - 1
   \]
   
   So the equation becomes:
   
   \[
   2\cos^2(\theta) - 1 = \cos(\theta)
   \]

3. **Rearrange the equation to form a quadratic equation:**

   \[
   2\cos^2(\theta) - \cos(\theta) - 1 = 0
   \]

4. **Solve the quadratic equation:**

   Let \(x = \cos(\theta)\), then the equation becomes:
   
   \[
   2x^2 - x - 1 = 0
   \]

   Use the quadratic formula:
   
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   
   Where \(a = 2\), \(b = -1\), and \(c = -1\).
   
   Substituting these values:
   
   \[
   x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)}
   \]
   
   Simplifying:
   
   \[
   x = \frac{1 \pm \sqrt{1 + 8}}{4} = \frac{1 \pm \sqrt{9}}{4} = \frac{1 \pm 3}{4}
   \]
   
   Therefore:
   
   \[
   x = 1 \quad \text{or} \quad x = -\frac{1}{2}
   \]

5. **Find the corresponding values of \(\theta\):**

   - For \(x = 1\), \(\cos(\theta) = 1\), so \(\theta = 0 + 2n\pi\), where \(n\) is any integer.
   - For \(x = -\frac{1}{2}\), \(\cos(\theta) = -\frac{1}{2}\), so \(\theta = \frac{2\pi}{3} + 2n\pi\) or \(\theta = \frac{4\pi}{3} + 2n\pi\).

Thus, the general solutions are:
\[
\theta = 0 + 2n\pi, \quad \theta = \frac{2\pi}{3} + 2n\pi, \quad \text{and} \quad \theta = \frac{4\pi}{3} + 2n\pi
\]
where \(n\) is any integer.

---

Would you like further details on this solution, or do you have any other questions?

### Related Questions
1. What is the general solution for \(\sin(2\theta) = \cos(\theta)\)?
2. How do you solve the equation \(\cos^2(\theta) = \frac{1}{2}\)?
3. What are the key identities to solve trigonometric equations?
4. How can you solve \(\tan(2\theta) = \sqrt{3}\) for \(\theta\)?
5. What is the difference between solving trigonometric equations and identities?
6. Can you derive the sum and difference formulas for sine and cosine?
7. What is the importance of the unit circle in solving trigonometric equations?
8. How do the double-angle and half-angle formulas help in solving trigonometric problems?

### Tip
Always start by simplifying trigonometric equations using known identities before attempting to solve them.

Learn how to solve the trigonometric equation cos(2theta) = sin(pi/2 - theta) step-by-step. This problem involves using the double-angle identity for cosine and solving a quadratic equation. Suitable for high school students studying trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation