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

Temperature ^oC
K
^oF

Length m
km
in
ft
yards
miles
naut miles

Area m²
km²
in²
ft²
miles²
acres

Volume m³
liters
in³
ft³
us gal

Weight kg_f
N
lbf

Velocity m/s
km/h
ft/min
ft/s
mph
knots

Pressure Pa
bar
mm H₂O
kg/cm²
psi
inches H₂O

Flow m³/s
m³/h
US gpm
cfm