Ximpledu

Trigonometry Formula

See how to use the trigonometry formulas
(trigonometric identities).
23 examples and their solutions.

sin θcos θ

Formula

sin θcos θ = tan θ

Example

Show that the given equation is true.

tan θsec2 θ = sin θ cos θ
Solution

sin2 θ + cos2 θ

Formula

sin2 θ + cos2 θ = 1
1 - sin2 θ = cos2 θ
1 - cos2 θ = sin2 θ
Move sin2 θ or cos2 θ to the right side.
tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
÷ sin2 θ or cos2 θ both sides.

Example

Show that the given equation is true.

(sin θ + cos θ)2sin θ = csc θ + 2 cos θ
Solution

Example

Simplify the given expression.

cos2 θ1 - sin θ
Solution

Example

Show that the given equation is true.

sin2 θ (1 + tan2 θ) = tan2 θ
Solution

Example

cos2 x - sin x + 1 = 0 (0 ≤ x ≤ 2π)
Solution

Trigonometry of (-θ)

Formula

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

Trigonometry of (90° - θ)

Formula

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

cos (A - B)

Formula

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

Example

cos 15°
Solution

cos (A + B)

Formula

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

Example

cos 75°
Solution

sin (A - B)

Formula

sin (A - B)
= sin A cos B - cos A sin B

Example

sin 15°
Solution

sin (A + B)

Formula

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

Example

sin 105°
Solution

Example

y = sin x + √3 cos x
Amplitude?
Solution

tan (A - B)

Formula

tan (A - B) = tan A - tan B1 + tan A tan B

Example

tan 15°
Solution

Example

tan θ = ?

Solution

tan (A + B)

Formula

tan (A + B) = tan A + tan B1 - tan A tan B

Example

tan 105°
Solution

Example

m = ?

Solution

sin 2θ

Formula

sin 2θ = 2 sin θ cos θ

Example

sin θ = 35, π2 ≤ θ ≤ π
sin 2θ = ?
Solution

Example

sin 2x = sin x (0 ≤ θ ≤ 2π)
Solution

cos 2θ

Formula

cos 2θ = 2 cos2 θ - 1
= cos2 θ - sin2 θ
= 1 - 2 sin2 θ
Use cos2 θ = 1 - sin2 θ
to find the middle & bottom formulas.

Example

cos θ = 14, cos 2θ = ?
Solution

Example

sin θ = -23, cos 2θ = ?
Solution

tan 2θ

Formula

tan 2θ = 2 tan θ1 - tan2 θ

Example

cos θ = -35, π2 ≤ θ ≤ π
tan 2θ = ?
Solution

Example

m = ?

Solution

sin θ2

Formula

sin θ2 = ±1 - cos θ2

Example

sin θ = -35, 2 ≤ θ ≤ 2π
sin θ2 = ?
Solution

cos θ2

Formula

cos θ2 = ±1 + cos θ2

Example

tan θ = -43, 2 ≤ θ ≤ 2π
cos θ2 = ?
Solution

tan θ2

Formula

cos θ2 = ±1 - cos θ1 + cos θ

Example

tan θ = 34, π ≤ θ ≤ 2
tan θ2 = ?
Solution