If `0 lt theta lt90^@, and sin (2theta + 50^@) = cos (4 theta + 16^@)`, then what s the value of `theta` (in degrees) ?

Options

A. `8^@`
B. `10^@`
C. `12^@`
D. `4^@`

Answer

To solve the equation \( \sin(2\theta + 50^\circ) = \cos(4\theta + 16^\circ) \), we can use the co-function identity which states that \( \cos(x) = \sin(90^\circ - x) \).

### Step-by-Step Solution:

1. **Apply the Co-function Identity**:
\[ \cos(4\theta + 16^\circ) = \sin(90^\circ - (4\theta + 16^\circ)) \]
This simplifies to:
\[ \sin(2\theta + 50^\circ) = \sin(90^\circ - 4\theta - 16^\circ) \]
Which can be rewritten as:
\[ \sin(2\theta + 50^\circ) = \sin(74^\circ - 4\theta) \]

2. **Set the Angles Equal**:
Since the sine function is equal for angles that differ by \( n \cdot 180^\circ \) (where \( n \) is an integer), we can set up the equation:
\[ 2\theta + 50^\circ = 74^\circ - 4\theta \]
or
\[ 2\theta + 50^\circ = 180^\circ - (74^\circ - 4\theta) \]

3. **Solve the First Equation**:
Rearranging the first equation:
\[ 2\theta + 4\theta = 74^\circ - 50^\circ \]
\[ 6\theta = 24^\circ \]
\[ \theta = 4^\circ \]

4. **Check the Second Equation**:
For the second equation:
\[ 2\theta + 50^\circ = 180^\circ - 74^\circ + 4\theta \]
Simplifying gives:
\[ 2\theta + 50^\circ = 106^\circ + 4\theta \]
Rearranging:
\[ 50^\circ - 106^\circ = 4\theta - 2\theta \]
\[ -56^\circ = 2\theta \]
This gives a negative angle which is not valid in the range \( 0 < \theta < 90 \).

5. **Conclusion**:
The only valid solution is:
\[ \theta = 4^\circ \]

### Final Answer:
\[ \theta = 4^\circ \]