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Math Formulas: Trigonometry Identities Right-Triangle Definitions 1.

sin α =

Opposite Hypotenuse

2.

cos α =

Adjacent Hypotenuse

3.

tan α =

Opposite Adjacent

4.

csc α =

1 Hypotenuse = sin α Opposite

5.

sec α =

1 Hypotenuse = cos α Adjacent

6.

cot α =

1 Adjacent = tan α Opposite

Reduction Formulas 7.

sin(−x) = − sin(x)

8.

cos(−x) = cos(x)

9.

sin

10.

cos

11.

sin

12.

cos

π 2 π 2 π

2 π 2

 − x = cos(x)  − x = sin(x)  + x = cos(x)

 + x = − sin(x)

13.

sin(π − x) = sin(x)

14.

cos(π − x) = − cos(x)

15.

sin(π + x) = − sin(x)

16.

cos(π + x) = − cos(x)

Basic Identities 17.

sin2 x + cos2 x = 1

18.

tan2 x + 1 =

1 cos2 x

19.

cot2 x + 1 =

1 sin2 x

Sum and Difference Formulas 1

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20.

sin(α + β) = sin α · cos β + sin β · cos α

21.

sin(α − β) = sin α · cos β − sin β · cos α

22.

cos(α + β) = cos α · cos β − sin α · cos β

23.

cos(α − β) = cos α · cos β + sin α · cos β

24.

tan(α + β) =

tan α + tan β 1 − tan α · tan β

25.

tan(α − β) =

tan α − tan β 1 + tan α · tan β

Double Angle and Half Angle Formulas 26.

sin(2 α) = 2 · sin α · cos α

27.

cos(2 α) = cos2 α − sin2 α 2 tan α 1 − tan2 α r 1 − cos α =± 2 r 1 + cos α =± 2

28.

tan(2 α) =

29.

sin

α 2

30.

cos

α 2

31.

tan

32.

α 1 − cos α sin α = = 2 sin α 1 − cos α r α 1 + cos α tan = ± 2 1 − cos α

Other Useful Trig Formulas Law of sines 33.

sin α sin β sin γ = = α β γ

Law of cosines a2 = b2 + c2 − 2 · b · c · cos α 34.

b2 = a2 + c2 − 2 · a · c · cos β c2 = a2 + b2 − 2 · a · b · cos γ

Area of triangle 35.

A=

1 a b sin γ 2

2