For `A=133^(@), 2cos\ A/2`is equal to

Options

A. `-sqrt(1+sin A)-sqrt(1-sin A)`
B. `-sqrt(1+sin A)+sqrt(1-sin A)`
C. `sqrt(1+sin A)-sqrt(1-sin A)`
D. `sqrt(1+sin A)+sqrt(1-sin A)`

Explanation

To find the value of \( 2 \cos \frac{A}{2} \) where \( A = 130^\circ \), we will follow these steps:

### Step 1: Calculate \( \frac{A}{2} \)
Given \( A = 130^\circ \),
\[ \frac{A}{2} = \frac{130^\circ}{2} = 65^\circ \]

**Hint:** Remember that dividing the angle by 2 gives you the angle in degrees.

### Step 2: Use the identity for \( 2 \cos \frac{A}{2} \)
We can use the identity:
\[ 2 \cos \frac{A}{2} = \sqrt{1 + \sin A} + \sqrt{1 - \sin A} \]

**Hint:** This identity helps relate the cosine of half an angle to the sine of the angle.

### Step 3: Calculate \( \sin A \)
Now, we need to find \( \sin 130^\circ \). Using the sine function:
\[ \sin 130^\circ = \sin (180^\circ - 50^\circ) = \sin 50^\circ \]
Using a calculator or trigonometric tables, we find:
\[ \sin 50^\circ \approx 0.7660 \]

**Hint:** Remember that \( \sin(180^\circ - x) = \sin x \).

### Step 4: Substitute \( \sin A \) into the identity
Substituting \( \sin A \) into the identity:
\[ 2 \cos \frac{A}{2} = \sqrt{1 + \sin 130^\circ} + \sqrt{1 - \sin 130^\circ} \]
\[ = \sqrt{1 + 0.7660} + \sqrt{1 - 0.7660} \]
\[ = \sqrt{1.7660} + \sqrt{0.2340} \]

**Hint:** Ensure you compute the square roots accurately.

### Step 5: Calculate the square roots
Calculating the square roots:
\[ \sqrt{1.7660} \approx 1.3270 \]
\[ \sqrt{0.2340} \approx 0.4830 \]

**Hint:** Use a calculator for precise values of square roots.

### Step 6: Add the results
Now, add the two results together:
\[ 2 \cos \frac{A}{2} \approx 1.3270 + 0.4830 = 1.8100 \]

**Hint:** Make sure to sum the values correctly for the final answer.

### Final Result
Thus, the value of \( 2 \cos \frac{A}{2} \) is approximately:
\[ \boxed{1.8100} \]