Cofunction identities
Connect complementary angles and paired trigonometric functions.
Cofunction identities is one of 5 identities formulas in the trigonometry section of this library, and it is used at high school level.
Why cofunction identities works
The two acute angles of a right triangle add to 90 degrees, so one angle's complement is the other angle. The leg opposite the first angle is the leg next to the second one, which means the fraction that counts as sine for one angle is the very same fraction that counts as cosine for the other.
What each symbol means
$\theta$ is any angle for which the functions are defined.
Cofunction identities: when it holds
In degree mode replace $\pi/2$ with $90^\circ$.
When it stops applying
There is no failure inside the stated conditions; the identity holds for every angle, including obtuse and negative ones, because it comes from a reflection of the whole circle rather than from the triangle picture alone.
Cofunction identities: a worked example
$\sin30^\circ=\cos60^\circ=1/2$.
The mistake to avoid
What people do: Students copy π/2 straight from the book while their calculator is set to degrees.
Why it goes wrong: In degree mode the machine reads π/2 as roughly 1.571 degrees, not as a quarter turn, so the whole angle inside the function is wrong.
Do this instead: Match the unit to the mode. In degrees, sin(90° − 30°) gives 0.866, which is cos 30 degrees. Leaving π/2 in place gives sin(−28.43°) = −0.476, a value that is not even positive.
Cofunction identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\theta$ is any angle for which the functions are defined.
- Check the conditions before substituting. In degree mode replace $\pi/2$ with $90^\circ$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Identities
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where cofunction identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about cofunction identities
Does this only work for acute angles?
No. The right-triangle picture explains the acute case, but the identity is true everywhere. Try θ = 200 degrees: sin(90° − 200°) = sin(−110°) = −0.940, which is exactly cos 200 degrees.
Is that where the co- in cosine comes from?
Yes, cosine is short for complement's sine. Tangent and cotangent pair up the same way, as do secant and cosecant, so each co- function is its partner evaluated at the complementary angle.
How does this help me solve equations?
It lets you convert an equation that mixes sine and cosine into one that uses just one of them. Turning cos θ into sin(90° − θ) makes both sides comparable, so you can match the angles directly.
Do tangent and cotangent swap the same way?
They do: tan(90° − θ) = cot θ. That is easy to see in a right triangle, where the opposite and adjacent legs simply exchange roles when you switch which acute angle you are looking at.