How can we find the exact value of sin 255 using addition or subtraction formulas?

To find the exact value of sin 255 degrees, we can use addition or subtraction formulas from trigonometry. Specifically, we can express 255 degrees as a sum of two angles whose sine values we know. One convenient way to do this is to write:

sin(255) = sin(270 – 15)

Now we can apply the sine subtraction formula:

sin(a – b) = sin(a)cos(b) – cos(a)sin(b)

Here, let a = 270 degrees and b = 15 degrees:

• sin(270) = -1
• cos(270) = 0
• sin(15) = rac{
  oot{6}-
  oot{2}}{4}
• cos(15) = rac{
  oot{6}+
  oot{2}}{4}

Now, substituting these values into our formula gives us:

sin(255) = sin(270)cos(15) – cos(270)sin(15)

Thus, this becomes:

sin(255) = (-1) * rac{9
oot{6}+
oot{2}}{4} – 0 * rac{9
oot{6}-
oot{2}}{4}

Therefore, we simplify it to:

sin(255) = -rac{9
oot{6}+
oot{2}}{4}

So, the exact value of sin 255 degrees is:

– rac{9
oot{6}+
oot{2}}{4}