sin$$ 2\theta $$=1 solve for 0$$ \ll\theta\ll2\pi $$

To solve the equation $\sin 2\theta = 1$ for $0 < \theta < 2\pi$, follow these steps:

### Step 1: Find the general solution for $2\theta$
The sine function equals 1 at specific points:
\[
\sin 2\theta = 1 \implies 2\theta = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}
\]
where $k$ is any integer.

### Step 2: Solve for $\theta$
Divide both sides by 2 to solve for $\theta$:
\[
\theta = \frac{\frac{\pi}{2} + 2k\pi}{2} = \frac{\pi}{4} + k\pi
\]

### Step 3: Determine valid solutions within the given range
Now, we consider values of $\theta$ in the range $0 < \theta < 2\pi$. We test values of $k$:

1. For $k = 0$:
   \[
   \theta = \frac{\pi}{4}
   \]
2. For $k = 1$:
   \[
   \theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}
   \]
3. For $k = 2$:
   \[
   \theta = \frac{\pi}{4} + 2\pi = \frac{9\pi}{4}
   \]
   However, $\frac{9\pi}{4}$ is greater than $2\pi$, so it is not within the range.

### Step 4: List the solutions
The solutions for $\theta$ in the range $0 < \theta < 2\pi$ are:
\[
\theta = \frac{\pi}{4}, \frac{5\pi}{4}
\]

These are the values of $\theta$ that satisfy the equation.

Would you like more details or have any questions?

---

Here are 5 related questions:
1. What are the general solutions for $\sin x = -1$?
2. How does the range of the sine function affect the solutions of trigonometric equations?
3. Can you solve $\cos 2\theta = 1$ for the same interval?
4. How would the solution change if the equation was $\sin 2\theta = -1$?
5. What are the implications of solving $\sin 2\theta = \frac{1}{2}$ within the same interval?

**Tip:** Always check your solutions by substituting them back into the original equation to verify correctness.

Learn how to solve the trigonometric equation sin(2θ) = 1 within the range 0 < θ < 2π. This solution involves understanding key concepts in trigonometry, including the sine function and general solutions 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