What is the exact value of 4 times the sine of 15 degrees times the cosine of 15 degrees?

To find the exact value of 4 times the sine of 15 degrees times the cosine of 15 degrees, we can utilize a well-known trigonometric identity. The product of sine and cosine can be expressed as follows:

[

\sin(a) \cos(a) = \frac{1}{2} \sin(2a)

]

In this case, we take ( a = 15 ) degrees, meaning:

[

\sin(15^\circ) \cos(15^\circ) = \frac{1}{2} \sin(30^\circ)

]

Since the sine of 30 degrees is known to be ( \frac{1}{2} ):

[

\sin(30^\circ) = \frac{1}{2}

]

Therefore, we can substitute that back into our equation:

[

\sin(15^\circ) \cos(15^\circ) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}

]

Now, multiplying this by 4 gives us:

[

4 \times \left(\sin(15^\circ) \cos(15^\circ