Trigonometry

The unit circle and trigonometric functions

Sine, cosine and tangent as coordinates on a circle; radians; and the identities worth remembering.

About 10 minutes

Trigonometry starts with ratios of sides in a right triangle, but those ratios only make sense for angles between 0° and 90°. The unit circle extends sine and cosine to every angle and explains their signs, their symmetries and why they repeat.

From triangles to the circle

In a right triangle with an angle $\theta$ : \sin\theta is opposite over hypotenuse, \cos\theta is adjacent over hypotenuse, and \tan\theta is opposite over adjacent.

Now draw a circle of radius 1 centred at the origin and rotate anticlockwise from the positive x-axis through the angle $\theta$ . The point you reach has coordinates (\cos\theta, \sin\theta). That is the definition for any angle.

(x, y) = (\cos\theta, \sin\theta),\qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Radians

A radian is the angle that cuts off an arc equal in length to the radius. A full turn is $2\pi$ radians, so 180° = \pi rad. Radians are the natural unit in calculus: the formula \frac{d}{dx}\sin x = \cos x is only true when $x$ is in radians.

\text{radians} = \text{degrees} \times \frac{\pi}{180}

Values and identities to know

The angles 30°, 45° and 60° have exact values: \sin 30° = \frac12, \sin 45° = \frac{\sqrt2}{2}, \sin 60° = \frac{\sqrt3}{2}, with the cosines in the reverse order.

Because every point on the unit circle satisfies x^2 + y^2 = 1, sine and cosine obey the Pythagorean identity. Sine and cosine repeat every $2\pi$ ; tangent repeats every $\pi$ .

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

Worked examples

Find \sin 150° exactly.

150° is in the second quadrant; its reference angle is 180° - 150° = 30°.
Sine is positive in the second quadrant.
\sin 150° = \sin 30° = \frac12.

Answer: \frac{1}{2}

Given \sin\theta = \frac35 with $\theta$ in the first quadrant, find \cos\theta.

\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}.
Cosine is positive in the first quadrant, so \cos\theta = \frac45.

Answer: \frac{4}{5}

Common mistakes

Leaving a calculator in degree mode when the angle is in radians, or the reverse.
Writing \sin^{-1} x and meaning \frac{1}{\sin x}. The notation \sin^{-1} is the inverse function (arcsine); the reciprocal is \csc x.
Assuming \sin(a + b) = \sin a + \sin b.
Forgetting that \tan\theta is undefined where \cos\theta = 0.

Practice problems

1.Convert 60° to radians
2.\cos 180°
3.\tan 45°
4.\sin\frac{7\pi}{6}

Tools for this lesson

Scientific CalculatorEvaluate expressions with powers, roots, factorials, logarithms and trigonometry in degrees or radians, with memory and history.Angle CalculatorConvert between degrees, radians, gradians and turns, find reference, complementary and supplementary angles, and the angle between lines.Graphing CalculatorPlot and compare functions, implicit equations and inequalities with zoom, pan, coordinate tracking and PNG export.Triangle CalculatorSolve a triangle from SSS, SAS, ASA, AAS or RHS: sides, angles, area by Heron’s formula, heights, medians, bisectors and radii.

Next lesson: Mean, median and standard deviation

From triangles to the circle

In a right triangle with an angle

\theta

\sin\theta

\sin\theta is opposite over hypotenuse,

\cos\theta

\cos\theta is adjacent over hypotenuse, and

\tan\theta

\tan\theta is opposite over adjacent.

Now draw a circle of radius 1 centred at the origin and rotate anticlockwise from the positive x-axis through the angle

\theta

. The point you reach has coordinates

(\cos\theta, \sin\theta)

(\cos\theta, \sin\theta). That is the definition for any angle.

(x, y) = (\cos\theta, \sin\theta),\qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

(x, y) = (\cos\theta, \sin\theta),\qquad \tan\theta = \frac{\sin\theta}{\cos\theta}

Radians

A radian is the angle that cuts off an arc equal in length to the radius. A full turn is

2\pi

radians, so

180° = \pi

180° = \pi rad. Radians are the natural unit in calculus: the formula

\frac{d}{dx}\sin x = \cos x

\frac{d}{dx}\sin x = \cos x is only true when

x

is in radians.

\text{radians} = \text{degrees} \times \frac{\pi}{180}

\text{radians} = \text{degrees} \times \frac{\pi}{180}

Values and identities to know

The angles 30°, 45° and 60° have exact values:

\sin 30° = \frac12

\sin 30° = \frac12,

\sin 45° = \frac{\sqrt2}{2}

\sin 45° = \frac{\sqrt2}{2},

\sin 60° = \frac{\sqrt3}{2}

\sin 60° = \frac{\sqrt3}{2}, with the cosines in the reverse order.

Because every point on the unit circle satisfies

x^2 + y^2 = 1

x^2 + y^2 = 1, sine and cosine obey the Pythagorean identity. Sine and cosine repeat every

2\pi

; tangent repeats every

\pi

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

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

Worked examples

Find $\sin 150°$ \sin 150° exactly.

150° is in the second quadrant; its reference angle is $180° - 150° = 30°$ 180° - 150° = 30°.
Sine is positive in the second quadrant.
$\sin 150° = \sin 30° = \frac12$ \sin 150° = \sin 30° = \frac12.

Answer: $\frac{1}{2}$ \frac{1}{2}

Given $\sin\theta = \frac35$ \sin\theta = \frac35 with $\theta$ in the first quadrant, find $\cos\theta$ \cos\theta.

$\cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}$ \cos^2\theta = 1 - \frac{9}{25} = \frac{16}{25}.
Cosine is positive in the first quadrant, so $\cos\theta = \frac45$ \cos\theta = \frac45.

Answer: $\frac{4}{5}$ \frac{4}{5}

Common mistakes

Leaving a calculator in degree mode when the angle is in radians, or the reverse.

Writing

\sin^{-1} x

\sin^{-1} x and meaning

\frac{1}{\sin x}

\frac{1}{\sin x}. The notation

\sin^{-1}

\sin^{-1} is the inverse function (arcsine); the reciprocal is

\csc x

\csc x.

Assuming

\sin(a + b) = \sin a + \sin b

\sin(a + b) = \sin a + \sin b.

Forgetting that

\tan\theta

\tan\theta is undefined where

\cos\theta = 0

\cos\theta = 0.