Trigonometric Functions of (-Theta)

Trigonometric Functions of (-Theta)

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

Formulas

sin (-theta) = -sin theta, cos (-theta) = cos theta, tan (-theta) = -tan theta

sin (-θ) = -sin θ
cos (-θ) = cos θ
tan (-θ) = -tan θ

The signs of sine and tangent change.
But the sign of cosine doesn't change.

Proofs

Trigonometric Functions of (-theta): Proofs of the Formulas

Draw a right triangle like this.

Write the central angle as θ.

Draw a right triangle
whose central angle is -θ.

Its base is the same: x.
But its height is -y.
So the hypotenuse's endpoint is (x, -y).

See the left right triangle.

Sine: SOH.
So sin θ = y/r.

Then, see the 'right' right triangle.

sin (-θ) = -y/r
= -(sin θ)

So sin (-θ) = -sin θ.

See the left right triangle.

Cosine: CAH.
So cos θ = x/r.

Then, see the 'right' right triangle.

cos (-θ) = x/r
= cos θ

So cos (-θ) = cos θ.

See the left right triangle.

Tangent: TOA.
So tan θ = y/x.

Then, see the 'right' right triangle.

tan (-θ) = -y/x
= -(tan θ)

So tan (-θ) = -tan θ.