Trigonometry Angles--Pi/6
TrigonometryAnglesPi6Diagram
Construction of the angle pi/6=30 degrees produces a 30-60-90 triangle, which has angles theta=pi/6 and 2theta=pi/3. From the above diagram, write y=sintheta for the vertical leg, then the horizontal leg is given by
| x=sqrt(1-y^2)=sin(2theta) |
(1)
|
by the Pythagorean theorem. Now use the double-angle formula
| sin(2theta)=2sinthetacostheta |
(2)
|
to obtain
| sqrt(1-y^2)=2ysqrt(1-y^2), |
(3)
|
which can be solved for y=sintheta to yield
| sintheta=1/2. |
(4)
|
Filling in the rest of the trigonometric functions then gives
[画像:cos(pi/6)] = 1/2sqrt(3)
(5)
[画像:cot(pi/6)] = sqrt(3)
(6)
[画像:csc(pi/6)] = 2
(7)
[画像:sec(pi/6)] = 2/3sqrt(3)
(8)
[画像:sin(pi/6)] = 1/2
(9)
[画像:tan(pi/6)] = 1/3sqrt(3).
(10)
See also
30-60-90 Triangle, Hexagon, Hexagram, Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/3Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/6." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi6.html