UNIT 9 • GEOMETRY & TRIGONOMETRY

Unit Circle
Sin, Cos & Tan of Any Angle

On the unit circle (radius 1), the point at angle $\theta$ is $(\cos\theta, \sin\theta)$ . Its x-coordinate is the cosine, its y-coordinate is the sine, and $\tan\theta = \frac{\sin\theta}{\cos\theta}$ .

READING THE CIRCLERADIUS = 1

\begin{gathered} P(\theta) = (\cos\theta,\ \sin\theta) \\ \cos^2\theta + \sin^2\theta = 1 \\ \theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180^\circ} \end{gathered}

1. QuadrantFind which quarter the angle lands in (subtract 360° first if it is bigger).

2. Reference AngleThe acute angle to the x-axis:

180^\circ - \theta

\theta - 180^\circ

360^\circ - \theta

3. Sign by ASTCTake the value for the reference angle and give it the sign of that quadrant.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the exact values of $\sin 150^\circ$ , $\cos 150^\circ$ and $\tan 150^\circ$ , and write 150° in radians.

STEP 01 • QUADRANT AND REFERENCE ANGLE

150° is between 90° and 180°, so it is in quadrant II. Its reference angle is $180^\circ - 150^\circ = 30^\circ$ .

STEP 02 • VALUES FOR 30°, THEN THE SIGNS

In quadrant II only sine is positive (ASTC: Sin):

\sin 150^\circ = +\sin 30^\circ = \frac{1}{2}, \qquad \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}

\tan 150^\circ = \frac{\sin 150^\circ}{\cos 150^\circ} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}

STEP 03 • RADIANS

150^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{6}

FINAL CONCLUSION

The point is (−√3/2, 1/2)

P(150^\circ) = \left(-\frac{\sqrt{3}}{2},\ \frac{1}{2}\right), \quad 150^\circ = \frac{5\pi}{6}

Drag the point on the circle below to 150° and the same values appear.

LABORATORY

Interactive Unit Circle

LABORATORY

Unit Circle Explorer

Angle θ

Typed angle is in

Snap to special angles while dragging

\theta = 60^\circ = \frac{\pi}{3}\ \text{rad}

\sin\theta = \frac{\sqrt{3}}{2}, \quad \cos\theta = \frac{1}{2}

\tan\theta = \sqrt{3}

Quadrant I · reference angle 60° · positive here: All

Working for this angle

60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \approx 1.0472 \text{ rad}

\text{Quadrant I, reference angle } = 60^\circ

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

Special Angles Table

Degrees	Radians	sin θ	cos θ	tan θ
0°	$0$	$0$	$1$	$0$
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	$1$	$0$	$\text{undefined}$
120°	$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$
135°	$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$
150°	$\frac{5\pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
180°	$\pi$	$0$	$-1$	$0$
210°	$\frac{7\pi}{6}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
225°	$\frac{5\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$
240°	$\frac{4\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\sqrt{3}$
270°	$\frac{3\pi}{2}$	$-1$	$0$	$\text{undefined}$
300°	$\frac{5\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$
315°	$\frac{7\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$
330°	$\frac{11\pi}{6}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
360°	$2\pi$	$0$	$1$	$0$

Common Questions

What is the unit circle?

A circle of radius 1 centred at the origin. The point at angle

\theta

(measured anticlockwise from the positive x-axis) is

(\cos\theta, \sin\theta)

, which is how sine and cosine are defined for every angle, not just acute ones.

How do you convert degrees to radians?

Multiply by

\frac{\pi}{180^\circ}

60^\circ = \frac{\pi}{3}

. To go back, multiply radians by

\frac{180^\circ}{\pi}

. One radian is about 57.3°.

What does "All Students Take Calculus" (ASTC) mean?

It tells you which ratios are positive in each quadrant: All in I, Sin in II, Tan in III, Cos in IV. That fixes the sign; the reference angle gives the size.

Why is tan 90° undefined?

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

, and

\cos 90^\circ = 0

, so you would be dividing by zero. The same happens at 270°.

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UNIT 9 • GEOMETRY & TRIGONOMETRY

Unit Circle
Sin, Cos & Tan of Any Angle

READING THE CIRCLERADIUS = 1

\begin{gathered} P(\theta) = (\cos\theta,\ \sin\theta) \\ \cos^2\theta + \sin^2\theta = 1 \\ \theta_{\text{rad}} = \theta^\circ \times \frac{\pi}{180^\circ} \end{gathered}

1. QuadrantFind which quarter the angle lands in (subtract 360° first if it is bigger).

2. Reference AngleThe acute angle to the x-axis:

180^\circ - \theta

\theta - 180^\circ

360^\circ - \theta

3. Sign by ASTCTake the value for the reference angle and give it the sign of that quadrant.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find the exact values of $\sin 150^\circ$ , $\cos 150^\circ$ and $\tan 150^\circ$ , and write 150° in radians.

STEP 01 • QUADRANT AND REFERENCE ANGLE

150° is between 90° and 180°, so it is in quadrant II. Its reference angle is $180^\circ - 150^\circ = 30^\circ$ .

STEP 02 • VALUES FOR 30°, THEN THE SIGNS

In quadrant II only sine is positive (ASTC: Sin):

\sin 150^\circ = +\sin 30^\circ = \frac{1}{2}, \qquad \cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}

\tan 150^\circ = \frac{\sin 150^\circ}{\cos 150^\circ} = \frac{1/2}{-\sqrt{3}/2} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}

STEP 03 • RADIANS

150^\circ \times \frac{\pi}{180^\circ} = \frac{5\pi}{6}

FINAL CONCLUSION

The point is (−√3/2, 1/2)

P(150^\circ) = \left(-\frac{\sqrt{3}}{2},\ \frac{1}{2}\right), \quad 150^\circ = \frac{5\pi}{6}

Drag the point on the circle below to 150° and the same values appear.

LABORATORY

Interactive Unit Circle

LABORATORY

Unit Circle Explorer

Angle θ

Typed angle is in

Snap to special angles while dragging

\theta = 60^\circ = \frac{\pi}{3}\ \text{rad}

\sin\theta = \frac{\sqrt{3}}{2}, \quad \cos\theta = \frac{1}{2}

\tan\theta = \sqrt{3}

Quadrant I · reference angle 60° · positive here: All

Working for this angle

60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \approx 1.0472 \text{ rad}

\text{Quadrant I, reference angle } = 60^\circ

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

Special Angles Table

Degrees	Radians	sin θ	cos θ	tan θ
0°	$0$	$0$	$1$	$0$
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	$1$	$0$	$\text{undefined}$
120°	$\frac{2\pi}{3}$	$\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$-\sqrt{3}$
135°	$\frac{3\pi}{4}$	$\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$-1$
150°	$\frac{5\pi}{6}$	$\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
180°	$\pi$	$0$	$-1$	$0$
210°	$\frac{7\pi}{6}$	$-\frac{1}{2}$	$-\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
225°	$\frac{5\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$-\frac{\sqrt{2}}{2}$	$1$
240°	$\frac{4\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$-\frac{1}{2}$	$\sqrt{3}$
270°	$\frac{3\pi}{2}$	$-1$	$0$	$\text{undefined}$
300°	$\frac{5\pi}{3}$	$-\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$-\sqrt{3}$
315°	$\frac{7\pi}{4}$	$-\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$-1$
330°	$\frac{11\pi}{6}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{3}}{3}$
360°	$2\pi$	$0$	$1$	$0$

Common Questions

What is the unit circle?

A circle of radius 1 centred at the origin. The point at angle

\theta

(measured anticlockwise from the positive x-axis) is

(\cos\theta, \sin\theta)

, which is how sine and cosine are defined for every angle, not just acute ones.

How do you convert degrees to radians?

Multiply by

\frac{\pi}{180^\circ}

60^\circ = \frac{\pi}{3}

. To go back, multiply radians by

\frac{180^\circ}{\pi}

. One radian is about 57.3°.

What does "All Students Take Calculus" (ASTC) mean?

It tells you which ratios are positive in each quadrant: All in I, Sin in II, Tan in III, Cos in IV. That fixes the sign; the reference angle gives the size.

Why is tan 90° undefined?

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

, and

\cos 90^\circ = 0

, so you would be dividing by zero. The same happens at 270°.

PreviousTriangle Solver

Unit 9•43 of 47

NextCoordinate Geometry

Unit Circle Sin, Cos & Tan of Any Angle

Step-by-Step Worked Example

The point is (−√3/2, 1/2)

Interactive Unit Circle

Unit Circle Explorer

Working for this angle

Special Angles Table

Common Questions

NETZ Engine

Unit Circle Sin, Cos & Tan of Any Angle

Step-by-Step Worked Example

The point is (−√3/2, 1/2)

Interactive Unit Circle

Unit Circle Explorer

Working for this angle

Special Angles Table

Common Questions

Unit Circle
Sin, Cos & Tan of Any Angle

Unit Circle
Sin, Cos & Tan of Any Angle