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List of trigonometric identities - Wikipedia, the free encyclopedia
en.wikipedia.org/wiki/List_of_trigonometric_identities
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Notation
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Angles
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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
grads
:
1 full circle = 360 degrees = 2
radians = 400 grads.
The following table shows the conversions for some common angles:
Degrees
30°
60°
120°
150°
210°
240°
300°
330°
Radians
Grads
33⅓ grad 66⅔ grad 133⅓ grad 166⅔ grad 233⅓ grad 266⅔ grad 333⅓ grad 366⅔ grad
Degrees
45°
90°
135°
180°
225°
270°
315°
360°
Radians
Grads
50 grad
100 grad
150 grad
200 grad
250 grad
300 grad
350 grad
400 grad
Unless otherwise specified, all angles in this article are assumed to be in radians, but angles ending in a degree symbol (°) are in degrees. Per
Niven's theorem
multiples of 30° are the
only
rational
angles with rational sin/cos, which may account for their popularity in examples.
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Trigonometric functions
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The primary trigonometric functions are the
sine
and
cosine
of an angle. These are sometimes abbreviated sin(
θ
) and cos(
θ
), respectively, where
θ
is the angle, but the parentheses around
the angle are often omitted, e.g., sin
θ
and cos
θ
.
The
tangent
(tan) of an angle is the
ratio
of the sine to the cosine:
Finally, the
reciprocal functions
secant (sec), cosecant (csc), and cotangent (cot) are the reciprocals of the cosine, sine, and tangent:
These definitions are sometimes referred to as
ratio identities
.
Inverse functions
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Main article:
Inverse trigonometric functions
The inverse trigonometric functions are partial
inverse functions
for the trigonometric functions. For example, the inverse function for the sine, known as the
inverse sine
(sin
−1
)
or
arcsine
(arcsin or asin), satisfies
and
This article uses the notation below for inverse trigonometric functions:
Function
sin
cos
tan
sec
csc
cot
Inverse
arcsin arccos arctan arcsec arccsc arccot
Pythagorean identity
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The basic relationship between the sine and the cosine is the
Pythagorean trigonometric identity
:
where
cos
2
θ
means
(cos(
θ
))
2
and
sin
2
θ
means
(sin(
θ
))
2
.
This can be viewed as a version of the
Pythagorean theorem
, and follows from the equation
x
2
+
y
2
= 1
for the
unit circle
. This equation can be solved for either the sine or the cosine:
Related identities
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Dividing the Pythagorean identity through by either
cos
2
θ
or
sin
2
θ
yields two other identities:
Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other (
up to
a plus or minus sign):
Each trigonometric function in terms of the other five.
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in terms of