Exact trigonometric values
irrational number produced by taking the sine or cosine of a rational multiple of a full circle
In mathematics, the values of the trigonometric functions can be expressed approximately, as in
cos
(
π
/
4
)
≈
0.707
{\displaystyle \cos(\pi /4)\approx 0.707}
, or exactly, as in
cos
(
π
/
4
)
=
2
/
2
{\displaystyle \cos(\pi /4)={\sqrt {2}}/2}
. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.
Common angles
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45° (see below for proofs). In the table below, the label "Undefined" represents a ratio
1
:
0.
{\displaystyle 1:0.}
If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectively extended real numbers, these entries take the value
∞
{\displaystyle \infty }
(see division by zero).
For angles outside of this range, trigonometric values can be found by applying reflection and shift identities, such as
sin
(
π
2
−
θ
)
=
cos
(
θ
)
,
sin
(
2
π
+
θ
)
=
sin
(
π
−
θ
)
=
sin
(
θ
)
,
sin
(
π
+
θ
)
=
sin
(
−
θ
)
=
−
sin
(
θ
)
,
cos
(
2
π
+
θ
)
=
cos
(
−
θ
)
=
cos
(
θ
)
,
cos
(
π
+
θ
)
=
cos
(
π
−
θ
)
=
−
cos
(
θ
)
.
{\displaystyle {\begin{alignedat}{3}&&\sin({\tfrac {\pi }{2}}-\theta )&{}=\cos(\theta ),\\[5mu]&&\sin(2\pi +\theta )&{}=\sin(\pi -\theta )&&{}=\sin(\theta ),\quad &&\sin(\pi +\theta )&&{}=\sin(-\theta )&&{}=-\sin(\theta ),\\[5mu]&&\cos(2\pi +\theta )&{}=\cos(-\theta )&&{}=\cos(\theta ),\quad &&\cos(\pi +\theta )&&{}=\cos(\pi -\theta )&&{}=-\cos(\theta ).\end{alignedat}}}
For example, for any θ greater than π/4, to find its sine, follow these steps to instead operate within the range of 0 to π/4:
While it is greater than 2π, subtract 2π from it. Now try to find sin(θ).
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