Exact Values on the Unit Circle (radians)
Author: AA
Date: 30 April 2013

Investigate the exact trigonometric values by dragging the green dot around the unit circle.

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 \; \; \; \frac{\pi}{2}

 \;  \pi\; 

 \; \; \; \frac{3\pi}{2}

2\pi \;    \;

\frac{\pi}{6}

\frac{\pi}{4}

\frac{\pi}{3}

\frac{2\pi}{3}

\frac{5\pi}{6}

\frac{3\pi}{4}

\frac{7\pi}{6}

\frac{5\pi}{4}

\frac{4\pi}{3}

ff0000

0000ff

\frac{11\pi}{6}

\frac{7\pi}{4}

\frac{5\pi}{3}

The Unit Circle

O = Intersect(xAxis, yAxis)
E = Point(xAxis)
c_1 = Circle(O, E)
A_1 = (0.8660254037844386, 0.5)
A_2 = (-0.8660254037844386, 0.5)
A_3 = (-0.8660254037844386, -0.5)
A_4 = (0.8660254037844386, -0.5)
C_1 = (0.5, 0.8660254037844386)
C_4 = (0.5, -0.8660254037844386)
C_3 = (-0.5, -0.8660254037844386)
C_2 = (-0.5, 0.8660254037844386)
B_1 = (0.7071067811865475, 0.7071067811865475)
B_4 = (0.7071067811865475, -0.7071067811865475)
B_3 = (-0.7071067811865475, -0.7071067811865475)
B_2 = (-0.7071067811865475, 0.7071067811865475)
N = (0, 1)
W = (-1, 0)
S = (0, -1)
X = Point(c_1)
a = Segment(O, X)
b = Distance(X, A_1)
α = Angle(E, O, X)
d = false
f = Segment(A_1, (x(A_1), 0))
g = Segment(A_2, (x(A_2), 0))
h = Segment(B_2, B_1)
i = Segment(C_2, C_1)
u = Vector(O, X)
onAngle: round(round(α) / 30) ≟ round(α) / 30 ∨ round(round(α) / 45) ≟ round(α) / 45
v = Vector(X, (x(X), 0))
w = Vector(X, (0, y(X)))
A = Point(u)
e = Segment(A, C_2)
j = Segment(A, B_2)
k = Segment(A, A_2)
l = Segment(A, A_3)
m = Segment(A, B_3)
n = Segment(A, C_3)
p = Segment(A, C_4)
q = Segment(A, B_4)
r = Segment(A, A_4)
s_1 = Segment(A, A_1)
t = Segment(A, B_1)
a_1 = Segment(A, C_1)
text3: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{1}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{\sqrt{3}}{2}}\right) "
text1: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{\sqrt{3}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{1}{2}}\right) "
text5: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{1},\fgcolor{" + (LaTeX[yColour]) + "}{0}\right)"
text13: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{1}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{\sqrt{3}}{2}}\right) "
text15: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{1}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{\sqrt{3}}{2}}\right) "
text16: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{1}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{\sqrt{3}}{2}}\right) "
text17: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{0},\fgcolor{" + (LaTeX[yColour]) + "}{-1}\right)"
text20: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-1},\fgcolor{" + (LaTeX[yColour]) + "}{0}\right)"
text4: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{0},\fgcolor{" + (LaTeX[yColour]) + "}{1}\right)"
text6: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{\sqrt{3}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{1}{2}}\right) "
text7: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{\sqrt{3}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{1}{2}}\right) "
text8: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{\sqrt{3}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{1}{2}}\right) "
Angles: {"0", "\frac{\pi}{6}", "\frac{\pi}{4}", "\frac{\pi}{3}", "\frac{\pi}{2}", "\frac{2\pi}{3}", "\frac{3\pi}{4}", "\frac{5\pi}{6}", "\pi", "\frac{7\pi}{6}", "\frac{5\pi}{4}", "\frac{4\pi}{3}", "\frac{3\pi}{2}", "\frac{5\pi}{3}", "\frac{7\pi}{4}", "\frac{11\pi}{6}", "2\pi"}
angleIndex: round(α / ° / 22.5) + 1
cosValues: {"1", "\frac{\sqrt{3}}{2}", "\frac{\sqrt{2}}{2}", "\frac{1}{2}", "0", "-\frac{1}{2}", "-\frac{\sqrt{2}}{2}", "-\frac{\sqrt{3}}{2}", "-1", "-\frac{\sqrt{3}}{2}", "-\frac{\sqrt{2}}{2}", "-\frac{1}{2}", "0", "\frac{1}{2}", "\frac{\sqrt{2}}{2}", "\frac{\sqrt{3}}{2}", "1"}
sinValues: {"0", "\frac{1}{2}", "\frac{\sqrt{2}}{2}", "\frac{\sqrt{3}}{2}", "1", "\frac{\sqrt{3}}{2}", "\frac{\sqrt{2}}{2}", "\frac{1}{2}", "0", "-\frac{1}{2}", "-\frac{\sqrt{2}}{2}", "-\frac{\sqrt{3}}{2}", "-1", "-\frac{\sqrt{3}}{2}", "-\frac{\sqrt{2}}{2}", "-\frac{1}{2}", "0"}
poly1, a_2, B, b_1 = Polygon(X, (x(X), 0), A)
poly2, d_1, e_1, a_3 = Polygon((abs(x(X)), abs(y(X))), A, (abs(x(X)), 0))
o: round(round(α) / 30) ≟ round(α) / 30
thirtyAngle: round(α) ≟ 150° ∨ round(α) ≟ 210° ∨ round(α) ≟ 330°
fortyfiveAngle: round(α) ≟ 135° ∨ round(α) ≟ 225° ∨ round(α) ≟ 315°
sixtyAngle: round(α) ≟ 120° ∨ round(α) ≟ 240° ∨ round(α) ≟ 300°
text36: "sin(" + (LaTeX[α]) + ")=" + (LaTeX[Element[sinValues, angleIndex]]) + ""
text35: "cos( " + (LaTeX[α]) + ")=" + (LaTeX[Element[cosValues, angleIndex]]) + ""
text2: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{\sqrt{2}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{\sqrt{2}}{2}}\right) "
text9: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{\sqrt{2}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{\frac{\sqrt{2}}{2}}\right) "
text10: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{-\frac{\sqrt{2}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{\sqrt{2}}{2}}\right) "
text11: "\left(\fgcolor{" + (LaTeX[xColour]) + "}{\frac{\sqrt{2}}{2}},\fgcolor{" + (LaTeX[yColour]) + "}{-\frac{\sqrt{2}}{2}}\right) "