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Trigonometric Functions

Sine, cosine and tangent - the natural trigonometric functions.

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Triangle - natural 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

cos( θ d ) = b / c

= 1 / sec( θ d )

= sin(π / 2 - θ r )                       (2)

tan( θ d ) = a / b

= 1 / cot( θ d )

= sin( θ d ) / cos( θ d )

= cot(π / 2 - θ r )                         (3)

cot( θ d ) = 1 / tan( θ d )

= cos( θ d ) / sin( θ d )

= tan(π / 2 - θ r )                          (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) = cos 2 (a) - sin 2 (a)                        (8b)

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

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

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

sin 2 (a) + cos 2 (a) = 1                         (8f)

sec 2 (a) = tan 2 (a) + 1                         (8g)

Special Triangles

Special triangles - trigonometric functions

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

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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