Trigonometric Functions

Natural trigonometric functions are expressed as

sin(θ_d) = a / c

= 1 / csc(θ_d)

= cos(π/2 - θ_r)                              (1)

where

θ_d = angle in degrees

θ_r = angle in radians

a

c

cos(θ_d) = b / c

= 1 / sec(θ_d)

= sin(π/2 - θ_r)                               (2)

b

c

tan(θ_d) = a / b

= 1 / cot(θ_d)

= sin(θ_d) / cos(θ_d)

= cot(π/2 - θ_r )                                     (3)

a

b

cot(θ_d) = 1 / tan(θ_d)

= cos(θ_d) / sin(θ_d)

= tan(π/2 - θ_r)                          (4)

θ (degrees)

Trigonometric functions ranging 0 to 90 degrees are tabulated below:

• Trigonometric functions in pdf-format

Inverse functions

arcsin(a) = sin^-1(a)                                   (1a)

arccos(a) = cos^-1(a)                                   (2a)

arctan(a) = tan^-1(a)                                     (3a)

Addition Formula

sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)                             (5)

cos(a ± b) = cos(a) cos(b) ± sin(a) sin(b)                            (5b)

tan(a ± b) = (tan(a) ± tan(b)) / (1 ± tan(a) tan(b))                         (5c)

Sum and Difference Formula

sin(a) + sin(b) = 2 sin((a + b)/2) cos((a + b)/2)                          (6)

sin(a) - sin(b) = 2 cos((a + b)/2) sin((a - b)/2)                          (6b)

cos(a) + cos(b) = 2 cos((a + b)/2) cos((a - b)/2)                          (6c)

cos(a) - cos(b) = - 2 sin((a + b)/2) sin((a - b)/2)                         (6d)

tan(a) + tan(b) = sin(a + b) / (cos(a) cos(b))                       (6e)

tan(a) - tan(b) = sin(a - b) / (cos(a) cos(b))                    (6f)

Product Formula

2 sin(a) cos(b) = sin(a - b) + sin(a + b)                       (7)

2 sin(a) sin(b) = cos(a - b) - cos(a - b)                      (7b)

2 cos(a) cos(b) = cos(a - b) + cos(a + b)                      (7c)

Multiple Angle and Powers Formula

sin(2 a) = 2 sin(a)  cos(a)                        (8)

cos(2 a) = cos²(a) - sin²(a)                        (8b)

cos(2 a) = 2 cos²(a) - 1                          (8c)

cos(2 a) = 1 - 2 sin²(a)                        (8d)

tan(2  a) = 2 tan(a) / (1 - tan²(a))                     (8e)

sin²(a) + cos²(a) = 1                         (8f)

sec²(a) = tan²(a) + 1                         (8g)

Special Triangles

Trigonometric Values

sin(-θ_d ) = - sin(θ_d )                     (9a)

where

θ_d = angle in degrees

sin(90° + θ_d) = cos(θ_d)                    (9b)

sin(90° - θ_d) = cos(θ_d)                   (9c)

sin(180° + θ_d) = - sin(θ_d)                 (9d)

sin(180° - θ_d) = sin(θ_d)                   (9e)

sin(270° + θ_d) = - cos(θ_d)                   (9f)

sin(270° - θ_d) = - cos(θ_d)                (9g)

sin(360° + θ_d) = sin(θ_d)                 (9h)

sin(360° - θ_d) = - sin(θ_d)              (9h)

cos(-θ_d) = cos(θ_d)                       (10a)

cos(90° + θ_d) = - sin(θ_d)                 (10b)

cos(90° - θ_d) = sin(θ_d)                    (10c)

cos(180° + θ_d) = - cos(θ_d)                    (10d)

cos(180° - θ_d) = - cos(θ_d)                 (10e)

cos(270° + θ_d) = sin(θ_d)                (10f)

cos(270° - θ_d) = - sin(θ_d)                    (10g)

cos(360° + θ_d) = cos(θ_d)                    (10h)

cos(360° - θ_d) = cos(θ_d)                   (10h)

tan(-θ_d) = - tan(θ_d)                        (11a)

tan(90° + θ_d) = - cot(θ_d)                   (11b)

tan(90° - θ_d) = cot(θ_d)                        (11c)

tan(180° + θ_d) = tan(θ_d)                  (11d)

tan(180° - θ_d) = - tan(θ_d)                  (11e)

tan(270° + θ_d) = - cot(θ_d)                   (11f)

tan(270° - θ_d) = cot(θ_d)                    (11g)

tan(360° + θ_d) = tan(θ_d)                    (11h)

tan(360° - θ_d) = - tan(θ_d)                     (11h)

Trigometric Functions of Common Angles

	0°	30°	45°	60°	90°
Sin	0	1 / 2	√2 / 2	√3 / 2	1
Cos	1	√2 / 2	√2 / 2	1 / 2	0
Tan	0	√3 / 3	1	√3	∞
Cot	∞	√3	1	√3 / 3	0
Sec	1	2 √3 / 3	√2	2	∞
Cosec	∞	2	√2	2 √3 / 3	1