How this calculator works
The endpoint of a radius at angle θ has coordinates (cos θ, sin θ). Dividing the angle by full turns gives a coterminal angle between 0° and 360°. The coordinate signs identify the quadrant, while their absolute values give the reference angle to the horizontal axis. On an axis, this tool reports reference angle 0° or 90° as appropriate. The table keeps common radical values exact rather than replacing them with decimals. π-text mode accepts a bounded multiple-and-divisor syntax, not arbitrary code or algebra.
Formula
x=cos θ; y=sin θ; x²+y²=1; tan θ=y/x when x≠0. One full turn is 2π radians or 360°. Reference angle=atan2(|y|,|x|).
Worked example
At 5π/6 radians, the reference angle is π/6, the quadrant is II and the coordinates are (-√3/2, 1/2). At -π/2 the endpoint is (0,-1), on the negative y-axis.
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Frequently asked questions
Why are coordinates dimensionless?
The circle’s radius is one unit, so cosine and sine represent the coordinates divided by that radius. They are ratios.
Can I enter a negative π angle?
Yes. For example -3*pi/4 turns clockwise to quadrant III and is coterminal with 5π/4.
Can I type pi/0 or a general expression?
No. A zero divisor is invalid, and forms such as sin(pi/6) or pi+1 are outside the accepted syntax. Use a numeric angle for other values.