List of trigonometric identities
Cosines and sines around the unit circle
In mathematics, trigonometric identities are equalities that involve trigonometric functionsand are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identitiesinvolving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Notation
Angles
This article uses Greek letters such as alpha(α), beta (β), gamma (γ), and theta (θ) to represent angles. Several different units of angle measure are widely used, including degrees, radians, and gradians (gons):
1 full circle (turn) = 360 degrees = 2π radians = 400 gons.
The following table shows the conversions and values for some common angles:
Conversions of common angles
0
|
0°
|
0
|
0g
|
0
|
1
|
0
|
1
/
12
|
30°
|
π
/
6
|
33
1
/
3
g
|
1
/
2
|
√3
/
2
|
√3
/
3
|
1
/
8
|
45°
|
π
/
4
|
50g
|
√2
/
2
|
√2
/
2
|
1
|
1
/
6
|
60°
|
π
/
3
|
66
2
/
3
g
|
√3
/
2
|
1
/
2
|
√3
|
1
/
4
|
90°
|
π
/
2
|
100g
|
1
|
0
|
undef./∞
|
1
/
3
|
120°
|
2π
/
3
|
133
1
/
3
g
|
√3
/
2
|
−
1
/
2
|
−√3
|
3
/
8
|
135°
|
3π
/
4
|
150g
|
√2
/
2
|
−
√2
/
2
|
−1
|
5
/
12
|
150°
|
5π
/
6
|
166
2
/
3
g
|
1
/
2
|
−
√3
/
2
|
−
√3
/
3
|
1
/
2
|
180°
|
π
|
200g
|
0
|
−1
|
0
|
7
/
12
|
210°
|
7π
/
6
|
233
1
/
3
g
|
−
1
/
2
|
−
√3
/
2
|
√3
/
3
|
5
/
8
|
225°
|
5π
/
4
|
250g
|
−
√2
/
2
|
−
√2
/
2
|
1
|
2
/
3
|
240°
|
4π
/
3
|
266
2
/
3
g
|
−
√3
/
2
|
−
1
/
2
|
√3
|
3
/
4
|
270°
|
3π
/
2
|
300g
|
−1
|
0
|
undef./∞
|
5
/
6
|
300°
|
5π
/
3
|
333
1
/
3
g
|
−
√3
/
2
|
1
/
2
|
−√3
|
7
/
8
|
315°
|
7π
/
4
|
350g
|
−
√2
/
2
|
√2
/
2
|
−1
|
11
/
12
|
330°
|
11π
/
6
|
366
2
/
3
g
|
−
1
/
2
|
√3
/
2
|
−
√3
/
3
|
1
|
360°
|
2π
|
400g
|
0
|
1
|
0
|
Results for other angles can be found at Trigonometric constants expressed in real radicals.
All angles in this article are re-assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per Niven's theoremmultiples of 30° are the only angles that are a rational multiple of one degree and also have a rational sine or cosine, which may account for their popularity in examples.[1][2]
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