Angle x is one of the acute angles in a right triangle. If the measure of angles is `30^(@),` what is the value of `(sin x)^(2) + (cos x )^(2)` ?

Options

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

Answer

To solve the problem, we need to find the value of \( \sin^2 x + \cos^2 x \) where \( x = 30^\circ \).

### Step-by-Step Solution:

1. **Identify the angle**:
We know that \( x = 30^\circ \).

2. **Calculate \( \sin 30^\circ \)**:
From trigonometric values, we know:
\[ \sin 30^\circ = \frac{1}{2} \]

3. **Calculate \( \cos 30^\circ \)**:
Similarly, we know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]

4. **Square the sine value**:
Now, we calculate \( \sin^2 30^\circ \):
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

5. **Square the cosine value**:
Next, we calculate \( \cos^2 30^\circ \):
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

6. **Add the squared values**:
Now, we add \( \sin^2 30^\circ \) and \( \cos^2 30^\circ \):
\[ \sin^2 30^\circ + \cos^2 30^\circ = \frac{1}{4} + \frac{3}{4} = \frac{4}{4} = 1 \]

### Final Answer:
Thus, the value of \( \sin^2 x + \cos^2 x \) is \( 1 \).