Trigonometric Functions of (90 Degrees - Theta)

Trigonometric Functions of (90 Degrees - Theta)

How to solve sin (90 degrees - theta), cos (90 degrees - theta), and tan (90 degrees - theta): formulas and their proofs.

Formulas

sin (90 degrees - theta) = cos theta, cos (90 degrees - theta) = sin theta, tan (90 degrees - theta) = cot theta

sin (90º - θ) = cos θ
cos (90º - θ) = sin θ
tan (90º - θ) = cot θ

Proofs

Trigonometric Functions of (90 Degrees - Theta): Proofs of the Formulas

Draw a right triangle like this.

Write the central angle as θ.

To draw a right triangle
whose central angle is 90º - θ,
let's see how the terminal side of 90º - θ is formed.

90º - θ = -θ + 90º

So first think about the terminal side of -θ.
Its endpoint is (x, -y).

Trigonometric functions of (-θ)

Next, rotate the terminal side 90º counterclockwise.

Then the endpoint of the terminal side becomes
(-[-y], x) = (y, x).

Rotation of 90º counterclockwise

Then draw a right triangle
whose central angle is 90º - θ.

See the left right triangle.

Cosine: CAH.
So cos θ = x/r.

Then, see the 'right' right triangle.

Sine: SOH.
So sin (90º - θ) = x/r.
= cos θ.

So sin (90º - θ) = cos θ.

See the left right triangle.

Sine: SOH.
So sin θ = y/r.

Then, see the 'right' right triangle.

Cosine: CAH.
So cos (90º - θ) = y/r
= sin θ.

So cos (90º - θ) = sin θ.

See the left right triangle.

Cotangent is the reciprocal of tangent.

And tangent: TOA.

So cot θ = 1/tan θ
= x/y.

Then, see the 'right' right triangle.

tan (90º - θ) = x/y
= cot θ.

So tan (90º - θ) = cot θ.