Precalculus · Grade 12

Unit circle

Turn the angle and watch the point, its shadows and the sine wave move together. Exact values appear at the 16 special angles.

Read the lesson: The unit circle

Turn the angle θ

Drag the gold point around the circle, or select the circle and use the arrow keys: 15° a press, with Shift 1°. Page Up and Page Down make a quarter turn; Home goes back to 0° and End to 359°.

The graph of y = sin θ from 0 to 360°. The traced part is the point’s height as θ grew from 0 to 0°.

cos θ: the shadow on the x-axis
sin θ: the shadow on the y-axis, traced on the wave

What the point tells you

Angle

θ = 0°

= $0$ radians

The point (cos θ, sin θ)

cos θ =: $1$
sin θ =: $0$
tan θ =: $0$

Exact values. tan θ = sin θ ÷ cos θ.

Reference angle

None. The point is on the positive x-axis, so read cos θ and sin θ straight from it.

Where the point is

On the positive x-axis

cos θ: positive
sin θ: zero
tan θ: zero

On an axis one coordinate is 0.

All 16 special angles

Exact values on the unit circle. Undefined means cos θ is 0.
θ in degrees	θ in radians	cos θ	sin θ	tan θ	Where
0°	$0$	$1$	$0$	$0$	On the positive x-axis
30°	$\frac{π}{6}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{3}$	Quadrant I
45°	$\frac{π}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	Quadrant I
60°	$\frac{π}{3}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\sqrt{3}$	Quadrant I
90°	$\frac{π}{2}$	$0$	$1$	undefined	On the positive y-axis
120°	$\frac{2 π}{3}$	$- \frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$- \sqrt{3}$	Quadrant II
135°	$\frac{3 π}{4}$	$- \frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$- 1$	Quadrant II
150°	$\frac{5 π}{6}$	$- \frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$- \frac{\sqrt{3}}{3}$	Quadrant II
180°	$π$	$- 1$	$0$	$0$	On the negative x-axis
210°	$\frac{7 π}{6}$	$- \frac{\sqrt{3}}{2}$	$- \frac{1}{2}$	$\frac{\sqrt{3}}{3}$	Quadrant III
225°	$\frac{5 π}{4}$	$- \frac{\sqrt{2}}{2}$	$- \frac{\sqrt{2}}{2}$	$1$	Quadrant III
240°	$\frac{4 π}{3}$	$- \frac{1}{2}$	$- \frac{\sqrt{3}}{2}$	$\sqrt{3}$	Quadrant III
270°	$\frac{3 π}{2}$	$0$	$- 1$	undefined	On the negative y-axis
300°	$\frac{5 π}{3}$	$\frac{1}{2}$	$- \frac{\sqrt{3}}{2}$	$- \sqrt{3}$	Quadrant IV
315°	$\frac{7 π}{4}$	$\frac{\sqrt{2}}{2}$	$- \frac{\sqrt{2}}{2}$	$- 1$	Quadrant IV
330°	$\frac{11 π}{6}$	$\frac{\sqrt{3}}{2}$	$- \frac{1}{2}$	$- \frac{\sqrt{3}}{3}$	Quadrant IV

Turn the angle θ

Drag the gold point around the circle, or select the circle and use the arrow keys: 15° a press, with Shift 1°. Page Up and Page Down make a quarter turn; Home goes back to 0° and End to 359°.

The graph of y = sin θ from 0 to 360°. The traced part is the point’s height as θ grew from 0 to 0°.

cos θ: the shadow on the x-axis
sin θ: the shadow on the y-axis, traced on the wave

What the point tells you

Angle

θ = 0°

= $0$ radians

The point (cos θ, sin θ)

cos θ =: $1$
sin θ =: $0$
tan θ =: $0$

Exact values. tan θ = sin θ ÷ cos θ.

Reference angle

None. The point is on the positive x-axis, so read cos θ and sin θ straight from it.

Where the point is

On the positive x-axis

cos θ: positive
sin θ: zero
tan θ: zero

On an axis one coordinate is 0.

Exact values on the unit circle. Undefined means cos θ is 0.

θ in degrees

θ in radians

cos θ

sin θ

tan θ

Where

0°

0

1

0

0

On the positive x-axis

30°

\frac{π}{6}

\frac{\sqrt{3}}{2}

\frac{1}{2}

\frac{\sqrt{3}}{3}

Quadrant I

45°

\frac{π}{4}

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

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

1

Quadrant I

60°

\frac{π}{3}

\frac{1}{2}

\frac{\sqrt{3}}{2}

\sqrt{3}

Quadrant I

90°

\frac{π}{2}

0

1

undefined

On the positive y-axis

120°

\frac{2 π}{3}

- \frac{1}{2}

\frac{\sqrt{3}}{2}

- \sqrt{3}

Quadrant II

135°

\frac{3 π}{4}

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

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

- 1

Quadrant II

150°

\frac{5 π}{6}

- \frac{\sqrt{3}}{2}

\frac{1}{2}

- \frac{\sqrt{3}}{3}

Quadrant II

180°

π

- 1

0

0

On the negative x-axis

210°

\frac{7 π}{6}

- \frac{\sqrt{3}}{2}

- \frac{1}{2}

\frac{\sqrt{3}}{3}

Quadrant III

225°

\frac{5 π}{4}

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

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

1

Quadrant III

240°

\frac{4 π}{3}

- \frac{1}{2}

- \frac{\sqrt{3}}{2}

\sqrt{3}

Quadrant III

270°

\frac{3 π}{2}

0

- 1

undefined

On the negative y-axis

300°

\frac{5 π}{3}

\frac{1}{2}

- \frac{\sqrt{3}}{2}

- \sqrt{3}

Quadrant IV

315°

\frac{7 π}{4}

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

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

- 1

Quadrant IV

330°

\frac{11 π}{6}

\frac{\sqrt{3}}{2}

- \frac{1}{2}

- \frac{\sqrt{3}}{3}

Quadrant IV

Study calculator

Unit circle

Turn the angle θ

What the point tells you

Angle

The point (cos θ, sin θ)

Reference angle

Where the point is

Study calculator

Unit circle

Turn the angle θ

What the point tells you

Angle

The point (cos θ, sin θ)

Reference angle

Where the point is