If `sin5theta=cos20^(@)(0^(@)ltthetalt90^(@))` then value of `theta` is

Options

A. `4^(@)`
B. `22^(@)`
C. `10^(@)`
D. `14^(@)`

Answer

To solve the equation \( \sin(5\theta) = \cos(20^\circ) \) under the condition \( 0^\circ < \theta < 90^\circ \), we can follow these steps:

### Step 1: Use the co-function identity
We know that \( \cos(20^\circ) \) can be expressed in terms of sine:
\[ \cos(20^\circ) = \sin(90^\circ - 20^\circ) = \sin(70^\circ) \]
Thus, we can rewrite the equation as:
\[ \sin(5\theta) = \sin(70^\circ) \]

### Step 2: Set up the sine equation
Since \( \sin(A) = \sin(B) \) implies that:
\[ A = B + n \cdot 360^\circ \quad \text{or} \quad A = 180^\circ - B + n \cdot 360^\circ \quad (n \in \mathbb{Z}) \]
we can apply this to our equation:
1. \( 5\theta = 70^\circ + n \cdot 360^\circ \)
2. \( 5\theta = 180^\circ - 70^\circ + n \cdot 360^\circ \)

### Step 3: Solve the first equation
For the first equation:
\[ 5\theta = 70^\circ \implies \theta = \frac{70^\circ}{5} = 14^\circ \]

### Step 4: Solve the second equation
For the second equation:
\[ 5\theta = 110^\circ + n \cdot 360^\circ \]
If we set \( n = 0 \):
\[ 5\theta = 110^\circ \implies \theta = \frac{110^\circ}{5} = 22^\circ \]
However, since \( n \) must be an integer and \( \theta \) must be less than \( 90^\circ \), we check for higher values of \( n \) but find that they exceed \( 90^\circ \).

### Step 5: Conclusion
Thus, the only valid solution within the given range is:
\[ \theta = 14^\circ \]

### Final Answer
The value of \( \theta \) is \( 14^\circ \).

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