Worked example

Find the exact value of sin 75°

Answer

(√6 + √2)/4

75° = 45° + 30°
Split into two special angles.
sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
Sine of a sum. · Sine of a sum
= (√2/2)(√3/2) + (√2/2)(1/2)
Special-angle values.
= √6/4 + √2/4 = (√6 + √2)/4
Combine.

sin(A + B) = sin A cos B + cos A sin B

Sine of a sum — The angle itself is a sum and you need the functions separately — to evaluate exactly (sin 75° = sin(45° + 30°)), to expand, or to recognise the pattern running backwards.

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.