What is the exact value of sin (15 degrees)?

The exact value of sin(15 degrees) can be found using the half-angle identity. Here are the steps:

1. We know that sin(30 degrees) = 1/2.

2. The half-angle identity for sine is sin(x/2) = ± sqrt[(1 - cos(x))/2].

3. Substituting 30 degrees into the half-angle identity gives sin(15 degrees) = sqrt[(1 - cos(30))/2].

4. We know that cos(30 degrees) = sqrt(3)/2.

5. Substituting sqrt(3)/2 into the equation gives sin(15 degrees) = sqrt[(1 - sqrt(3)/2)/2].

6. Simplifying the equation gives sin(15 degrees) = sqrt[2 - sqrt(3)]/2.

So, the exact value of sin(15 degrees) is sqrt[2 - sqrt(3)]/2.

The exact value of sin(15 degrees) can be found using the half-angle identity, which states that sin(θ/2) = ± sqrt((1 - cos(θ))/2).

Here's the step-by-step calculation:

1. Convert 15 degrees to radians because the trigonometric functions in mathematics are typically defined in terms of radians, not degrees. To convert degrees to radians, multiply by π/180. So, 15 degrees is 15π/180 = π/12 radians.

2. Use the half-angle identity to find sin(π/12). But we don't know cos(π/6) yet.

3. We know that cos(π/6) = sqrt(3)/2 from the unit circle definition of cosine.

4. Substitute cos(π/6) into the half-angle formula: sin(π/12) = sqrt((1 - sqrt(3)/2)/2).

5. Simplify the expression under the square root: (1 - sqrt(3)/2)/2 = (2 - sqrt(3))/4.

6. Take the square root of the simplified expression: sin(π/12) = sqrt((2 - sqrt(3))/4) = sqrt(2 - sqrt(3))/2.

So, the exact value of sin(15 degrees) is sqrt(2 - sqrt(3))/2.