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Unit circle

Unit circle
θ is an angle measure.
sin(θ) is the y-coordinate.
cos(θ) is the x-coordinate.
a is the arc length in radians.

The unit circle is a special circle of geometry with a radius of 1 and its center positioned in the point (0,0) in an ordinary coordinate system.

The trigonometric functions, like sine, cosine and tangent, can among other things be defined and illustrated graphically with the unit circle.

The unit circle is not very complicated, and if you understand the theory behind it, it’s much easier to understand calculations with sine and cosine.

sine, cosine and tangent

If A is a point on the unit circle, where the radius touches the circle, and θ is the angle between the x-axis and the radius (counterclockwise turning is positive), then cos(θ) is the x-coordinate of the point A and sin(θ) is the y-coordinate of the point A. Tangent is the slope of the radius and therefore defined as the relationship between sin(θ) and cos(θ):

tan(theta)=frac{sin(theta)}{cos(theta)}

Try the interactive figure to the right.

You can find tangent on the unit circle, if you make a straight vertical line passing through the point (1,0), also called a tangent to the circle, and extent the radius line passing through A(cos(θ), sin(θ)), so that it touches the vertical circle’s tangent as well. The point where the two lines touches each other is (1, tan(θ)).

Radians

An angle θ in radians is the arc length of the unit circle from point (1,0) to point (cos(θ), sin(θ)).

Other relations of trigonometric functions on the unit circle

x^2+y^2=1

cos^2(theta)+sin^2(v)=1

cos(theta)=cos(2*n*k+theta)
for any integer k.

sin(theta)=sin(2*n*k+theta)
for any integer k.

Important angles in radians and in degrees

Important angles appearing on the unit circle

DegreesRadians(X,Y) coordinates

0^0

0

(1,0)

30^0

frac{pi}{6}

(frac{sqrt{3}}{2},frac{1}{2})

45^0

frac{pi}{4}

(frac{sqrt{2}}{2},frac{sqrt{2}}{2})

60^0

frac{pi}{3}

(frac{1}{2},frac{sqrt{3}}{2})

90^0

frac{pi}{2}

(0,1)

120^0

frac{2*pi}{3}

(frac{-1}{2},frac{sqrt{3}}{2})

135^0

frac{3*pi}{4}

(frac{-sqrt{2}}{2},frac{sqrt{2}}{2})

150^0

frac{5*pi}{6}

(frac{-sqrt{3}}{2},frac{1}{2})

180^0

pi

(-1,0)

210^0

frac{7*pi}{6}

(frac{-sqrt{3}}{2},frac{-1}{2})

225^0

frac{5*pi}{4}

(frac{-sqrt{2}}{2},frac{-sqrt{2}}{2})

240^0

frac{4*pi}{3}

(frac{-1}{2},frac{-sqrt{3}}{2})

270^0

frac{3*pi}{2}

(0,-1)

300^0

frac{5*pi}{3}

(frac{1}{2},frac{-sqrt{3}}{2})

315^0

frac{7*pi}{4}

(frac{sqrt{2}}{2},frac{-sqrt{2}}{2})

330^0

frac{11*pi}{6}

(frac{sqrt{3}}{2},frac{-1}{2})