Worked example

If sin 32° = 0.53, what is cos 58°?

Answer

cos 58° = 0.53 — the two angles are complementary.

32° + 58° = 90°
Check whether the angles are complementary — that is what makes cofunction useful.
cos 58° = sin(90° − 58°)
Cofunction identity for cosine. · Cofunction identity for cosine
= sin 32°
Subtract.
= 0.53
Given value.

sin θ = cos(90° − θ)

cos θ = sin(90° − θ)

Cofunction identity for sine — When the two angles in a problem add to 90° (π/2), the sine of one is the cosine of the other. Typical case: sin 32° asked as a cosine, or a calculus substitution that needs cos written as a shifted sine.
Cofunction identity for cosine — The reverse direction of the sine cofunction: use it to move a cosine into a sine (or back) when only one of them is integrable or listed in a table.

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.