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About this tool
Move around the unit circle to see the coordinates, quadrant, reference angle and all six trig ratios for any angle.
The Unit Circle Explorer shows where any angle lands on the circle of radius 1 centred at the origin, giving the exact point (cos θ, sin θ) along with its quadrant, reference angle and all six trigonometric ratios. It works from the definition of the unit circle itself — x = cos θ and y = sin θ, with tan θ = y/x — and prints the standard surd values at the sixteen special angles that are multiples of 30° and 45°. It is built for trigonometry and precalculus students who need to see why the signs flip from quadrant to quadrant.
Open Unit Circle Explorer on AltFTool — it loads instantly in your browser.
Enter the values you already know.
Fine-tune the options to match your scenario.
Read the result and use it in your planning or reporting.
Shows √3/2 and √2/2 forms at every multiple of 30° and 45°, not just decimals.
Where cos θ or sin θ is zero, the tool prints 'undefined' instead of a huge number.
Gives the acute angle to the x-axis so you can reuse first-quadrant values in any quadrant.
They are (cos θ, sin θ). At 30° the point is (√3/2, 1/2), which is about (0.8660, 0.5000), because the circle has radius 1 and the x and y legs of the reference triangle are exactly the cosine and sine.
Because tan θ = sin θ / cos θ and cos 90° = 0, so the division has no value. The same happens at 270°; cotangent and cosecant are undefined instead at 0° and 180°, where sin θ = 0.
A reference angle is the acute angle between the terminal side and the x-axis, always between 0° and 90°. Subtract from 180° in quadrant II, subtract 180° in quadrant III, and subtract from 360° in quadrant IV — so 210° has a reference angle of 30°.
All six are positive in quadrant I, only sine and cosecant in quadrant II, only tangent and cotangent in quadrant III, and only cosine and secant in quadrant IV — the 'All Students Take Calculus' mnemonic.
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