Unit Circle Explorer

Drag the point around the circle or type an angle. See all six trig values update in real time, with exact forms for standard angles.

Angle°

Point on Circle

\left(\frac{\sqrt{2}}{2},\; \frac{\sqrt{2}}{2}\right)

(0.7071, 0.7071)

Angle

Degrees

45°

Radians

\frac{\pi}{4}

Reference Angle

45°

Quadrant

I

Signs in Quadrant I

sin +cos +tan +csc +sec +cot +

Trigonometric Values

sin θ+

\frac{\sqrt{2}}{2}

csc θ+

\sqrt{2}

cos θ+

\frac{\sqrt{2}}{2}

sec θ+

\sqrt{2}

tan θ+

1

cot θ+

1

Exact values shown for this standard angle.

Open this tool with your own values

This tool reads its inputs from the page address, so a link can open it with the values from a problem already entered. Include every value the problem needs, because one left out keeps the tool's default. Use plain numbers in the stated units.

a: angle in degrees

Example link

https://livephysics.com/tools/unit-circle-explorer/?a=210

Reference Guide

What is the Unit Circle?

The unit circle is a circle with radius 1, centered at the origin of a coordinate plane. Any point on the circle can be written as $(\cos\theta,\; \sin\theta)$ where $\theta$ is the angle measured counterclockwise from the positive x-axis.

x^2 + y^2 = 1

This connects angle measurement directly to coordinate geometry. Every trigonometric function can be defined in terms of the unit circle.

CAST Rule (Signs by Quadrant)

Q II

Sin +

Q I

All +

Q III

Tan +

Q IV

Cos +

The mnemonic "All Students Take Calculus" names the positive function going counterclockwise from Q I. The reciprocal function shares the same sign as its primary.

Standard Angle Exact Values

Angle	0°	30°	45°	60°	90°
sin	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
cos	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
tan	$0$	$\frac{\sqrt{3}}{3}$	$1$	$\sqrt{3}$	undef

Values for angles beyond 90° can be found using the reference angle and the CAST rule for signs.

Fundamental Trig Identities

Pythagorean

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

1 + \tan^2\theta = \sec^2\theta

1 + \cot^2\theta = \csc^2\theta

Reciprocal

\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

Quotient

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}

Geometric Meaning

Toggle the colored lines on the circle to see what each trig function represents geometrically.

Sine. The vertical distance from the x-axis up (or down) to the point on the circle.

Cosine. The horizontal distance from the origin to where the point projects onto the x-axis.

Tangent. The length of the segment along the vertical tangent line at (1, 0), from the x-axis to where the extended radius meets it.

TopicsTrigonometry Functions Graphs Shapes Mathematics

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