Sin, Cos and Tan
Created | Updated Jun 30, 2003
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These are functions for finding the ratio of the sides in a right-angeld triangle.
Sin (sine)
The ratio of the Opposite and Hypotenuse sides.
Cos (cosine)
The ratio of the Adjacent and Hypotenuse sides.
Tan (tangent)
The ratio of the Opposite and Adjacent sides.
Values
Angle (θ) | Degrees | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
Radians | 0r | π/6r | π/4r | π/3r | π/2r | 4π/6r | 3π/4r | 5π/6r | πr | |
Sin θ | 0 | 1 / 2 | √(1/2) | √3 / 2 | 1 | √3 / 2 | √(1/2) | 1 / 2 | 0 | |
Cos θ | 1 | √3 / 2 | √(1/2) | 1 / 2 | 0 | -1 / 2 | -√(1/2) | -√3 / 2 | -1 | |
Tan θ | 0 | 1 / √3 | 1 / 2 | √3 | ∞ | -√3 | -1 / 2 | -1 / √3 | 0 | |
Angle (θ) | Degrees | 180° | 210° | 225° | 240° | 270° | 300° | 315° | 345° | 360° |
Radians | πr | 7π/6r | 5π/4r | 4π/3r | 3π/2r | 10π/6r | 7π/4r | 11π/6r | 2πr | |
Sin θ | 0 | -1 / 2 | -√(1/2) | -√3 / 2 | -1 | -√3 / 2 | -√(1/2) | -1 / 2 | 0 | |
Cos θ | -1 | -√3 / 2 | -√(1/2) | -1 / 2 | 0 | 1 / 2 | √(1/2) | √3 / 2 | 1 | |
Tan θ | 0 | 1 / √3 | 1 / 2 | √3 | ∞ | -√3 | -1 / 2 | -1 / √3 | 0 | |
If you draw a graph of cos or sin, it will form a wave-like pattern called a sinusoidal curve.
Sec, Cosec and Cot
Sec, Cosec and Cot are used instead of 1÷Cos, 1÷Sin and 1÷Tan.
sec x = | 1 |
cos x | |
cosec x = | 1 |
sin x | |
cot x = | 1 |
tan x |
Arccos, Arcsin and Arctan
These are the opposites of sin, cos and tan. For example, sin 30° = 1/2, so arcsin 1/2 = 30°.
These functions are usually represented by sin-1 x, invsin x or asin x on your calculator,
and may also be called inverse sine/cosine/tangent.
Notation
Sin, cos and tan can be written with or without brackets.
sin θ is the same as sin(θ)
Powers can be written in the middle, like this:
sin2θ means (sin θ)2
You should not use sin-1θ. This is to stop confusion between arcsin(θ) and 1/(sin θ) (which is cosec θ)
Conversion
Conversion between cos, sin and tan must be done using equations like the ones below.
sin2A + cos2A = 1
sin(A±B) = sin(A)cos(B) ± cos(A)sin(B)
cos(A±B) = cos(A)cos(B) ± -sin(A)sin(B)
tan(A±B) = {tan(A)A ± Btan(B)} / {1 ± -tan(A)tan(B)}
sin(2A) = 2sin(A)cos(A)
cos(2A) = cos2(A) - sin2(A)
tan(2A) = 2tan(A) / {1 - tan2(A)}
sin2(½A) = ½{1 - cos(A)}
cos2(½A) = ½{1 + cos(A)}
