Compute
`sin 15 ^(@) ` from the functions `60^(@) and 45 ^(@)`

Answer

To compute \( \sin 15^\circ \) using the values of \( \sin 60^\circ \) and \( \sin 45^\circ \), we can use the sine subtraction formula. Here’s how we can do it step by step:

### Step 1: Identify the angles
We know that:
\[ 15^\circ = 60^\circ - 45^\circ \]

### Step 2: Use the sine subtraction formula
The sine subtraction formula states:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
In our case, let \( a = 60^\circ \) and \( b = 45^\circ \). Therefore:
\[ \sin 15^\circ = \sin(60^\circ - 45^\circ) = \sin 60^\circ \cos 45^\circ - \cos 60^\circ \sin 45^\circ \]

### Step 3: Substitute known values
We know the following trigonometric values:
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 60^\circ = \frac{1}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)

Substituting these values into the formula gives:
\[ \sin 15^\circ = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{1}{\sqrt{2}}\right) - \left(\frac{1}{2}\right) \left(\frac{1}{\sqrt{2}}\right) \]

### Step 4: Simplify the expression
Now, we simplify the expression:
\[ \sin 15^\circ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \]
To combine these fractions, we need a common denominator:
\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Final Result
Thus, the value of \( \sin 15^\circ \) is:
\[ \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

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