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Trigonometric Functions

Sine, cosine and tangent - the natural trigonometric functions.

Triangle - natural trigonometric functions

Natural trigonometric functions are expressed as

sin(θd) = a / c

= 1 / csc(θd)

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

where

θd = angle in degrees

θr = angle in radians

cos(θd) = b / c

= 1 / sec(θd)

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

tan(θd) = a / b

= 1 / cot(θd)

= sin(θd) / cos(θd)

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

cot(θd) = 1 / tan(θd)

= cos(θd) / sin(θd)

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

Trigonometric functions ranging 0 to 90 degrees are tabulated below:

Trigonometric functions - sine cosine tangent

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

Special triangles - trigonometric functions

Trigonometric Values

sin(-θd ) = - sin(θd )                     (9a)

where

θd = angle in degrees

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

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

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

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

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

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

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

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

cos(-θd) = cos(θd)                       (10a)

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

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

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

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

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

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

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

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

tan(-θd) = - tan(θd)                        (11a)

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

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

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

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

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

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

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

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

Trigometric Functions of Common Angles

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
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5.16.11

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