To find the exact value of sin 255°, we can utilize the sine addition formula. First, let’s express 255° in terms of angles for which we know the sine values. We can rewrite 255° as:
255° = 180° + 75°
Now, we can apply the sine addition formula:
sin(A + B) = sin A cos B + cos A sin B
Here, A = 180° and B = 75°.
From trigonometric values, we know:
- sin 180° = 0
- cos 180° = -1
- sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
- (From known values: sin 45° = √2/2, cos 30° = √3/2, cos 45° = √2/2, sin 30° = 1/2)
- sin 75° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
- cos 75° = cos(45° + 30°) = cos 45° cos 30° – sin 45° sin 30°
- cos 75° = (√2/2)(√3/2) – (√2/2)(1/2) = (√6 – √2)/4
Now substituting the values back into our formula, we have:
sin 255° = sin(180° + 75°) = sin 180° cos 75° + cos 180° sin 75°
Plugging in the values:
sin 255° = 0 * cos 75° + (-1) * sin 75°
sin 255° = 0 – (√6 + √2)/4 = -(√6 + √2)/4
Thus, the exact value of sin 255° is:
sin 255° = - rac{√6 + √2}{4}