# Natural Trigonometric Functions

## Sine, Cosine and Tangent - Natural Trigonometric Functions

Natural trigonometric functions are expressed as

sin (θd) = a / c

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

where

θd = angle in degrees

cos (θd) = b / c

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

tan (θd) = a / b

= sin (θd) / cos(θd)

= 1 / cot (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:

### Inverse functions

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

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

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

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)

## Related Topics

• Basics - The SI-system, unit converters, physical constants, drawing scales and more
• Mathematics - Mathematical rules and laws - numbers, areas, volumes, exponents, trigonometric functions and more

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