In `(0, 2pi)`, the number of solutions of `cos2theta=sintheta` are

Options

A. `1`
B. `2`
C. `3`
D. `4`

Answer

To solve the equation \( \cos 2\theta = \sin \theta \) in the interval \( (0, 2\pi) \), we will follow these steps:

### Step 1: Use Trigonometric Identity
We start by using the double angle identity for cosine:
\[ \cos 2\theta = 1 - 2\sin^2 \theta \]
Thus, we can rewrite the equation:
\[ 1 - 2\sin^2 \theta = \sin \theta \]

**Hint:** Remember the double angle identities for sine and cosine, which can help simplify the equation.

### Step 2: Rearrange the Equation
Next, we rearrange the equation to one side:
\[ 1 - 2\sin^2 \theta - \sin \theta = 0 \]
This can be rewritten as:
\[ 2\sin^2 \theta + \sin \theta - 1 = 0 \]

**Hint:** Rearranging the equation helps in forming a standard quadratic equation.

### Step 3: Factor the Quadratic Equation
Now, we will factor the quadratic equation:
\[ (2\sin \theta - 1)(\sin \theta + 1) = 0 \]

**Hint:** Look for two numbers that multiply to give the product of the leading coefficient and the constant term, and add to give the middle coefficient.

### Step 4: Solve Each Factor
Now, we solve each factor separately:
1. \( 2\sin \theta - 1 = 0 \)
\[ \sin \theta = \frac{1}{2} \]
2. \( \sin \theta + 1 = 0 \)
\[ \sin \theta = -1 \]

**Hint:** Set each factor to zero to find the possible values of \( \sin \theta \).

### Step 5: Find Solutions in the Interval \( (0, 2\pi) \)
1. For \( \sin \theta = \frac{1}{2} \):
- The solutions are:
\[ \theta = \frac{\pi}{6}, \quad \frac{5\pi}{6} \]
2. For \( \sin \theta = -1 \):
- The solution is:
\[ \theta = \frac{3\pi}{2} \]

**Hint:** Remember the unit circle to determine the angles corresponding to the sine values.

### Step 6: Count the Solutions
Now we count the solutions we found:
- From \( \sin \theta = \frac{1}{2} \): \( \frac{\pi}{6}, \frac{5\pi}{6} \) (2 solutions)
- From \( \sin \theta = -1 \): \( \frac{3\pi}{2} \) (1 solution)

Thus, the total number of solutions in the interval \( (0, 2\pi) \) is:
\[ \text{Total Solutions} = 2 + 1 = 3 \]

**Final Answer:** The number of solutions of \( \cos 2\theta = \sin \theta \) in the interval \( (0, 2\pi) \) is **3**.