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# 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

cos 15°
Solution

## cos (A + B)

### Formula

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

cos 75°
Solution

## sin (A - B)

### Formula

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

sin 15°
Solution

## sin (A + B)

### Formula

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

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

tan 15°
Solution

tan θ = ?

Solution

## tan (A + B)

### Formula

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

tan 105°
Solution

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

m = ?

Solution

## sinθ2

### Formula

sin θ2 = ±1 - cos θ2

### Example

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

## cosθ2

### Formula

cos θ2 = ±1 + cos θ2

### Example

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

## tanθ2

### Formula

cos θ2 = ±1 - cos θ1 + cos θ

### Example

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