If `tantheta_(1)=1,sintheta_(2)=(1)/(sqrt(2))` then the value of `sin(theta_(1)+theta_(2))` equal to

Options

A. `-1`
B. 0
C. 1
D. `(1)/(2)`

Answer

To solve the problem, we need to find the value of \( \sin(\theta_1 + \theta_2) \) given that \( \tan(\theta_1) = 1 \) and \( \sin(\theta_2) = \frac{1}{\sqrt{2}} \).

### Step-by-Step Solution:

1. **Determine \( \theta_1 \)**:
- Given \( \tan(\theta_1) = 1 \).
- The angle \( \theta_1 \) for which \( \tan(\theta_1) = 1 \) is \( 45^\circ \) (or \( \frac{\pi}{4} \) radians).

2. **Determine \( \theta_2 \)**:
- Given \( \sin(\theta_2) = \frac{1}{\sqrt{2}} \).
- The angle \( \theta_2 \) for which \( \sin(\theta_2) = \frac{1}{\sqrt{2}} \) is also \( 45^\circ \) (or \( \frac{\pi}{4} \) radians).

3. **Calculate \( \theta_1 + \theta_2 \)**:
- Now, we have \( \theta_1 = 45^\circ \) and \( \theta_2 = 45^\circ \).
- Therefore, \( \theta_1 + \theta_2 = 45^\circ + 45^\circ = 90^\circ \).

4. **Find \( \sin(\theta_1 + \theta_2) \)**:
- We need to find \( \sin(90^\circ) \).
- The value of \( \sin(90^\circ) \) is \( 1 \).

### Final Answer:
Thus, the value of \( \sin(\theta_1 + \theta_2) \) is \( 1 \).

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