The value of `sin 50^(@) - sin 70^(@) + sin 10^(@)` is

Options

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

Answer

To find the value of \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \), we can use the sine subtraction identity.

### Step-by-Step Solution:

1. **Rewrite the expression**:
\[ \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \]

2. **Use the sine subtraction formula**:
The formula for the difference of sines is:
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
Here, let \( A = 50^\circ \) and \( B = 70^\circ \).

3. **Calculate \( A + B \) and \( A - B \)**:
\[ A + B = 50^\circ + 70^\circ = 120^\circ \]
\[ A - B = 50^\circ - 70^\circ = -20^\circ \]

4. **Apply the formula**:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos\left(\frac{120^\circ}{2}\right) \sin\left(\frac{-20^\circ}{2}\right) \]
\[ = 2 \cos(60^\circ) \sin(-10^\circ) \]

5. **Substitute the values of cosine and sine**:
We know that \( \cos(60^\circ) = \frac{1}{2} \) and \( \sin(-10^\circ) = -\sin(10^\circ) \):
\[ = 2 \cdot \frac{1}{2} \cdot (-\sin(10^\circ)) = -\sin(10^\circ) \]

6. **Combine with \( \sin 10^\circ \)**:
Now, substituting back into the original expression:
\[ -\sin(10^\circ) + \sin(10^\circ) = 0 \]

### Final Answer:
The value of \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \) is \( 0 \).

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