`cos15^(@)=?`

Options

A. `sqrt((1+cos30^(@))/(2))`
B. `sqrt((1-cos30^(@))/(2))`
C. `pmsqrt((1+cos30^(@))/(2))`
D. `pmsqrt((1-cos30^(@))/(2))`

Answer

To find the value of \( \cos 15^\circ \), we can use the cosine of a double angle identity and some known values of cosine. Here’s a step-by-step solution:

### Step 1: Use the Cosine Double Angle Identity
We know that:
\[ \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \]
We can also express this as:
\[ \cos(2\theta) = 2\cos^2(\theta) - 1 \]

### Step 2: Set \( \theta = 15^\circ \)
Let’s set \( \theta = 15^\circ \). Therefore, \( 2\theta = 30^\circ \).

### Step 3: Substitute into the Identity
Using the identity:
\[ \cos(30^\circ) = 2\cos^2(15^\circ) - 1 \]
We know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \).

### Step 4: Set Up the Equation
Now we can set up the equation:
\[ \frac{\sqrt{3}}{2} = 2\cos^2(15^\circ) - 1 \]

### Step 5: Solve for \( \cos^2(15^\circ) \)
Rearranging the equation gives:
\[ 2\cos^2(15^\circ) = \frac{\sqrt{3}}{2} + 1 \]
\[ 2\cos^2(15^\circ) = \frac{\sqrt{3}}{2} + \frac{2}{2} = \frac{\sqrt{3} + 2}{2} \]
Now divide both sides by 2:
\[ \cos^2(15^\circ) = \frac{\sqrt{3} + 2}{4} \]

### Step 6: Take the Square Root
Now, take the square root to find \( \cos(15^\circ) \):
\[ \cos(15^\circ) = \sqrt{\frac{\sqrt{3} + 2}{4}} = \frac{\sqrt{\sqrt{3} + 2}}{2} \]

### Final Answer
Thus, the value of \( \cos 15^\circ \) is:
\[ \cos(15^\circ) = \frac{\sqrt{\sqrt{3} + 2}}{2} \]

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