Use the sum and difference identities to determine the exact value of the following expression.sin(255°)

Sure, let's use the sum and difference identities to find the exact value of sin(255°).

First, we need to express 255° as a sum or difference of angles that we know the sine of. The most common angles are 30°, 45°, 60°, 90°, 180°, etc.

255° can be expressed as 180° + 75°.

Now, we can use the sum identity for sine, which is sin(a + b) = sin(a)cos(b) + cos(a)sin(b).

Substituting a = 180° and b = 75° into the formula, we get:

sin(255°) = sin(180° + 75°) = sin(180°)cos(75°) + cos(180°)sin(75°)

We know that sin(180°) = 0 and cos(180°) = -1.

So, sin(255°) = 0cos(75°) - 1sin(75°) = -sin(75°)

Now, we can express 75° as 45° + 30° and use the sum identity again:

-sin(75°) = -[sin(45°)cos(30°) + cos(45°)sin(30°)]

We know that sin(45°) = cos(45°) = √2/2 and sin(30°) = 1/2, cos(30°) = √3/2.

So, -sin(75°) = -[(√2/2 * √3/2) + (√2/2 * 1/2)] = -[√6/4 + √2/4] = -[√6 + √2]/4

Therefore, sin(255°) = -[√6 + √2]/4.