Unit Circle Calculator

About the Unit Circle Calculator

The unit circle turns trigonometry into coordinates. Every angle corresponds to a point on a circle of radius 1, and the cosine and sine of the angle are simply that point's $x$ - and $y$ -coordinates. Memorising a handful of special points, then using symmetry, gives exact values for dozens of angles without a calculator.

This calculator does that reasoning for you. It converts the angle between degrees and radians, reduces it to one turn, finds the quadrant and the reference angle, and reads off exact values such as $\frac{\sqrt3}{2}$ for the special angles. For other angles it gives accurate decimals. It also reports tangent, secant, cosecant and cotangent, and says when one of them is undefined.

What is the unit circle?

The unit circle is the circle $x^2 + y^2 = 1$ , centred at the origin with radius 1. An angle $\theta$ is measured from the positive $x$ -axis, anticlockwise for positive angles and clockwise for negative ones. The point where its terminal side meets the circle is $(\cos\theta, \sin\theta)$ .

Because that point lies on the circle, $\cos^2\theta + \sin^2\theta = 1$ for every angle, the Pythagorean identity. Tangent is the ratio $\frac{\sin\theta}{\cos\theta}$ , the slope of the terminal side, and the reciprocal functions are $\sec\theta = \frac{1}{\cos\theta}$ , $\csc\theta = \frac{1}{\sin\theta}$ and $\cot\theta = \frac{\cos\theta}{\sin\theta}$ .

How to use the Unit Circle Calculator

Enter the angle, for example 150, -30 or 5pi/4.
Choose degrees or radians.
Press Find values.
Read the quadrant, the reference angle and the exact coordinates.
Use the tangent and reciprocal values listed below the coordinates.

Formulas

Point on the circle

P(\theta) = (\cos\theta,\ \sin\theta)

Degrees to radians

\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}

Pythagorean identity

\cos^2\theta + \sin^2\theta = 1

Tangent

\tan\theta = \frac{\sin\theta}{\cos\theta}

Worked example: θ = 150°

The example uses $\theta = 150°$ . The angle lies in Quadrant II, where $x$ is negative and $y$ is positive, and its reference angle is $180° - 150° = 30°$ .

The point for $30°$ is $(\frac{\sqrt3}{2}, \frac12)$ . Applying the Quadrant II signs gives $(-\frac{\sqrt3}{2}, \frac12)$ , and dividing sine by cosine gives $\tan 150° = -\frac{\sqrt3}{3}$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)

How we read your problem

Unit circle values for 150°

\theta = 150^\circ

Step-by-step solution

Convert degrees to radians (multiply by π/180)
$\theta = 150^\circ = \frac{5\pi}{6}\ \text{rad}$
Locate the terminal side: in Quadrant II
In Quadrant II sine is positive and cosine is negative (“All Students Take Calculus”).
Reference angle (acute angle to the x-axis)
$\theta_{\text{ref}} = 30^\circ$
Read the exact coordinates from the reference angle and apply the quadrant signs
$(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$
Tangent = sine ÷ cosine
$\tan\theta = \frac{\sin\theta}{\cos\theta} = -\frac{\sqrt{3}}{3}$
Reciprocal functions
$\csc\theta = 2,\quad \sec\theta = -\frac{2\sqrt{3}}{3},\quad \cot\theta = -\sqrt{3}$

Formulas used

Point on the unit circle
$(x, y) = (\cos\theta, \sin\theta)$
Tangent and reciprocals
$\tan\theta = \tfrac{\sin\theta}{\cos\theta},\ \csc\theta = \tfrac{1}{\sin\theta},\ \sec\theta = \tfrac{1}{\cos\theta},\ \cot\theta = \tfrac{\cos\theta}{\sin\theta}$
Degrees and radians
$\text{rad} = \text{deg} \times \frac{\pi}{180}$

150° (5π/6 rad) ends in Quadrant II with a reference angle of 30°. sin θ = 1/2, cos θ = −√3/2, tan θ = −√3/3, csc θ = 2, sec θ = −2√3/3, cot θ = −√3.

Verification

cos²θ + sin²θ = 1 (must equal 1).

Detailed explanation

Special angles

Three acute angles have simple exact values. At $30°$ the point is $(\frac{\sqrt3}{2}, \frac12)$ , at $45°$ it is $(\frac{\sqrt2}{2}, \frac{\sqrt2}{2})$ , and at $60°$ it is $(\frac12, \frac{\sqrt3}{2})$ . They come from the 30-60-90 and 45-45-90 triangles. Together with the axis angles $0°$ , $90°$ , $180°$ and $270°$ , they generate the familiar sixteen points of the unit circle.

Reference angles and quadrant signs

Every angle has a reference angle: the acute angle between its terminal side and the $x$ -axis. The coordinates of the angle have the same size as those of its reference angle; only the signs change with the quadrant. In Quadrant I both are positive, in II only sine is positive, in III only tangent is positive, and in IV only cosine is positive. So $\frac{5\pi}{4}$ , which is $225°$ in Quadrant III with reference angle $45°$ , gives $(-\frac{\sqrt2}{2}, -\frac{\sqrt2}{2})$ .

Angles larger than a full turn or negative angles are first reduced by adding or subtracting $360°$ . The calculator reduces $765°$ to $45°$ , and treats $-30°$ as $330°$ in Quadrant IV, where the point is $(\frac{\sqrt3}{2}, -\frac12)$ .

Undefined values and non-special angles

Tangent and secant divide by the cosine, so they are undefined where the terminal side is vertical, at $90°$ and $270°$ . Cotangent and cosecant divide by the sine and are undefined at $0°$ and $180°$ . The calculator states these cases instead of printing a meaningless number.

Most angles, such as $20°$ , have no simple radical form. For these the calculator gives decimal values, still using the quadrant and reference angle so you can see why the signs come out as they do.

Common mistakes

Mixing up the coordinates: cosine is the $x$ -coordinate and sine is the $y$ -coordinate.
Using degree mode for an angle written with $\pi$ , or radian mode for an angle in degrees.
Measuring the reference angle from the $y$ -axis instead of the $x$ -axis.
Applying the wrong quadrant sign; remember which functions are positive in each quadrant.
Writing $\tan 90°$ as a large number instead of undefined.

Tips

Learn the values for $30°$ , $45°$ and $60°$ ; everything else follows by symmetry.
Count in multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$ to place radian angles quickly.
Rationalise when needed: $\frac{1}{\sqrt3} = \frac{\sqrt3}{3}$ .
Check results with $\cos^2\theta + \sin^2\theta = 1$ .
Use the trigonometric identity calculator to prove relationships between these functions.

Frequently asked questions

What are the coordinates of a point on the unit circle?

For an angle θ the point is (cos θ, sin θ). The x-coordinate is the cosine and the y-coordinate is the sine.

How do I find a reference angle?

Reduce the angle to between 0° and 360°, then measure the acute angle to the x-axis: θ in Quadrant I, 180° − θ in II, θ − 180° in III and 360° − θ in IV.

Can I enter angles in radians?

Yes. Choose radians and type expressions such as pi/3 or 5pi/4. The calculator converts between radians and degrees in the steps.

Why is tan 90° undefined?

Tangent is sine divided by cosine, and cos 90° = 0. Division by zero is undefined, so tan 90° and sec 90° have no value.

Does the calculator handle negative angles and angles over 360°?

Yes. It reduces them to an equivalent angle within one turn. For example 765° reduces to 45° and −30° is treated as 330°.

What happens with angles that are not special?

For angles like 20° there is no simple exact form, so the calculator shows decimal values while still explaining the quadrant and reference angle.

Which trigonometric functions are positive in each quadrant?

All are positive in Quadrant I, sine and cosecant in II, tangent and cotangent in III, and cosine and secant in IV.

What does All Students Take Calculus mean?

It is a mnemonic for the quadrant signs, read anticlockwise from Quadrant I: All functions are positive in I, Sine in II, Tangent in III and Cosine in IV. The calculator quotes it in the quadrant step so you can connect each sign to its quadrant.

Find exact trig values

Enter any angle above to see its unit-circle point, reference angle and all six trigonometric values.

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