Trigonometry Angles--Pi/3
TrigonometryAnglesPi3
Construction of the angle pi/3=60 degrees produces a 30-60-90 triangle, which has angles theta=pi/3 and theta/2=pi/6. From the above diagram, write y=sintheta for the vertical leg, then the horizontal leg is given by
| x=sqrt(1-y^2)=sin(1/2theta) |
(1)
|
by the Pythagorean theorem. Now use the double-angle formula
| sintheta=2sin(1/2theta)cos(1/2theta) |
(2)
|
to obtain
| y=2sqrt(1-y^2)y, |
(3)
|
which can be solved for y=sintheta to yield
| sintheta=1/2sqrt(3). |
(4)
|
Filling in the remainder of the trigonometric functions then gives
[画像:cos(pi/3)] = 1/2
(5)
[画像:cot(pi/3)] = 1/3sqrt(3)
(6)
[画像:csc(pi/3)] = 2/3sqrt(3)
(7)
[画像:sec(pi/3)] = 2
(8)
[画像:sin(pi/3)] = 1/2sqrt(3)
(9)
[画像:tan(pi/3)] = sqrt(3).
(10)
See also
30-60-90 Triangle, Equilateral Triangle, Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/6Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/3." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi3.html