1.7.1 Angles, the unit circle, sine and cosine

Degrees and radians

An angle measures an amount of turning. A full turn is 360 degrees (360°), so a half turn is 180° and a quarter turn, a right angle, is 90°. Turning anticlockwise counts as positive and turning clockwise as negative.

Maths and Python use a second unit, the radian, which comes from a length. Draw the unit circle: radius 1, centred at the origin $(0, 0)$ . An angle of $\theta$ radians at the centre ( $\theta$ is the Greek letter theta) cuts off an arc of length $\theta$ on that circle.

One radian cuts off an arc as long as the radius, about 57.3°. Six radians and a bit go round once: a full turn is $2\pi \approx 6.28$ radians, and a half turn is $\pi$ .

The whole circle has length $2\pi \approx 6.283185$ , so a full turn is $2\pi$ radians. That single fact gives every conversion: $360° = 2\pi$ , so $180° = \pi$ and one radian is $\frac{180}{\pi} \approx 57.2958$ degrees.

On a circle of radius $r$ , every arc is $r$ times as long, so the arc length is $r\theta$ with $\theta$ in radians. A radian is a length divided by a length, so it has no units. An angle written without the ° sign is in radians.

Rule: Converting angles

\begin{gathered} \text{radians} = \text{degrees} \times \frac{\pi}{180} \\ \text{degrees} = \text{radians} \times \frac{180}{\pi} \end{gathered}

Both come from $180° = \pi$ radians. The angles below are worth knowing in both units.

degrees	radians	as a decimal
30°	$\frac{\pi}{6}$	0.523599
45°	$\frac{\pi}{4}$	0.785398
60°	$\frac{\pi}{3}$	1.047198
90°	$\frac{\pi}{2}$	1.570796
180°	$\pi$	3.141593
270°	$\frac{3\pi}{2}$	4.712389
360°	$2\pi$	6.283185

Going back, $\frac{3\pi}{4}$ radians is $\frac{3}{4}$ of 180°, which is 135°.

\frac{3\pi}{4}

import math for degrees in [30, 45, 60, 90, 180, 270, 360]: print(degrees, round(math.radians(degrees), 6)) print(math.degrees(3 * math.pi / 4)) print(round(math.degrees(1), 4))

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Sine and cosine on the unit circle

Start at the point $(1, 0)$ on the unit circle and turn through an angle $\theta$ . The across coordinate of the point you reach is the cosine of $\theta$ , written $\cos\theta$ . The up coordinate is the sine, $\sin\theta$ .

Rule: Sine and cosine

\text{point at angle } \theta = (\cos\theta,\ \sin\theta)

Cosine is the horizontal coordinate and sine the vertical one. The point stays on a circle of radius 1, so both lie between $-1$ and 1.

The quarter turns land on the axes. At 0 the point is $(1, 0)$ , so $\cos 0 = 1$ and $\sin 0 = 0$ . A quarter turn reaches $(0, 1)$ , a half turn $(-1, 0)$ , and three quarters $(0, -1)$ : so $\sin\frac{\pi}{2} = 1$ and $\cos\pi = -1$ .

Sine and cosine are functions in the sense of lesson 1.1.1: each angle gives one output. Their domain is every real number and their range is $[-1, 1]$ .

Move the slider and watch the two coordinates. Then try the presets 30°, 150°, 210° and 330°: the sine is 0.5 or $-0.5$ each time. Switch on radians to see each angle as a fraction of $\pi$ .

Explore

Turn the angle. The point on the circle is (cos θ, sin θ): its across distance is the cosine and its height is the sine.

Angle θ in degrees = 30

Show radians

Angle: θ = 30°
cos θ (across): 0.866025
sin θ (up): 0.5
Where: quadrant 1; cos² θ + sin² θ = 1

Exact values at 30°, 45° and 60°

Three angles come up so often that their exact values are worth knowing. Each comes from a triangle and Pythagoras' theorem. If a right-angled triangle has shorter sides $a$ and $b$ and longest side $c$ , then $a^2 + b^2 = c^2$ .

45°. The radius at 45° is the long side of a right-angled triangle with two equal legs, so $\cos 45° = \sin 45°$ . Call that value $a$ . Then $a^2 + a^2 = 1$ , so $a^2 = \frac{1}{2}$ and $a = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707107$ .

60°. The point at 60°, the origin and $(1, 0)$ make a triangle. Two of its sides are radii, with 60° between them, so all three sides are 1. The point sits above the middle of the base, so $\cos 60° = \frac{1}{2}$ , and Pythagoras gives $\sin 60° = \sqrt{1 - \frac{1}{4}} = \frac{\sqrt{3}}{2} \approx 0.866025$ .

30°. The point at 30° is the point at 60° mirrored in the line $y = x$ , so the two coordinates swap: $\cos 30° = \frac{\sqrt{3}}{2}$ and $\sin 30° = \frac{1}{2}$ .

The legs of each triangle are the cosine and sine of its angle. At 30° the two legs of the 60° triangle swap places.

angle	0	30° = $\frac{\pi}{6}$	45° = $\frac{\pi}{4}$	60° = $\frac{\pi}{3}$	90° = $\frac{\pi}{2}$
$\cos\theta$	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
$\sin\theta$	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1

A pattern helps to remember the sine row: it is $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$ . The cosine row is the same list backwards.

Try it Each line prints the angle, then cosine and sine from math, then the exact values from the table as decimals. The two pairs agree.

import math exact = {30: (math.sqrt(3) / 2, 1 / 2), 45: (math.sqrt(2) / 2, math.sqrt(2) / 2), 60: (1 / 2, math.sqrt(3) / 2)} for degrees, (c, s) in exact.items(): t = math.radians(degrees) print(degrees, round(math.cos(t), 6), round(math.sin(t), 6), round(c, 6), round(s, 6))

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Signs, and every other angle

The two axes cut the plane into four quadrants, numbered I to IV anticlockwise from the top right. Inside each one the signs of the two coordinates are fixed.

quadrant	angles	$\cos\theta$	$\sin\theta$
I	0° to 90°	$+$	$+$
II	90° to 180°	$-$	$+$
III	180° to 270°	$-$	$-$
IV	270° to 360°	$+$	$-$

Every point on the circle is a mirror image of one in quadrant I. Its reference angle is the angle between its radius and the horizontal axis, always between 0° and 90°.

Rule: Any angle

\begin{gathered} \text{size from the reference angle} \\ \text{sign from the quadrant} \end{gathered}

The coordinates have the sizes of the reference angle's coordinates. The quadrant decides which of them are negative.

Example 1. Find the exact cosine and sine of 150°, 225° and 300°.

150° is 30° short of 180°, in quadrant II. The reference angle is 30° and the cosine is negative: $\cos 150° = -\frac{\sqrt{3}}{2}$ , $\sin 150° = \frac{1}{2}$ .
225° is 45° past 180°, in quadrant III. Both are negative: $\cos 225° = \sin 225° = -\frac{\sqrt{2}}{2} \approx -0.707107$ .
300° is 60° short of 360°, in quadrant IV. Only the sine is negative: $\cos 300° = \frac{1}{2}$ , $\sin 300° = -\frac{\sqrt{3}}{2}$ .

A negative angle turns clockwise, which mirrors the point in the horizontal axis. The across part stays and the up part flips: $\cos(-\theta) = \cos\theta$ and $\sin(-\theta) = -\sin\theta$ . So $-30°$ lands where 330° does, at $\left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$ .

-30°

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The Pythagorean identity

The point $(\cos\theta, \sin\theta)$ is always 1 from the origin. Its across and up distances are the legs of a right-angled triangle with long side 1, so Pythagoras' theorem holds at every angle.

Rule: Pythagorean identity

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

$\cos^2\theta$ means $(\cos\theta)^2$ . Knowing one of the two and the quadrant gives the other.

Example 2. An angle $\theta$ in quadrant IV has $\cos\theta = 0.6$ . Find $\sin\theta$ .

The identity gives $\sin^2\theta = 1 - 0.36 = 0.64$ , so $\sin\theta$ is 0.8 or $-0.8$ . In quadrant IV the sine is negative, so $\sin\theta = -0.8$ .

\cos^2 + \sin^2

import math for degrees in [0, 30, 150, 225, 1000]: t = math.radians(degrees) print(degrees, round(math.cos(t)**2 + math.sin(t)**2, 12)) theta = -math.acos(0.6) print(round(theta, 6), round(math.cos(theta), 6), round(math.sin(theta), 6))

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Periodic functions and their graphs

One more full turn brings the point back to the same place. So sine and cosine repeat every $2\pi$ , that is every 360°. A function that repeats like this is periodic, and the length of the shortest repeat is its period.

Rule: Period of sine and cosine

\begin{gathered} \sin(\theta + 2\pi) = \sin\theta \\ \cos(\theta + 2\pi) = \cos\theta \end{gathered}

Adding or taking away any whole number of turns changes neither. So 30°, 390°, 750° and $-330°$ all have sine $\frac{1}{2}$ .

Plotted against $\theta$ , both make the same wave between $-1$ and 1. Sine is 0 at every multiple of $\pi$ , and cosine is 0 halfway between, at $\frac{\pi}{2}$ , $\frac{3\pi}{2}$ and so on. The cosine wave is the sine wave moved $\frac{\pi}{2}$ to the left, a shift from lesson 1.1.2: $\cos\theta = \sin\left(\theta + \frac{\pi}{2}\right)$ .

Tangent

The tangent of $\theta$ is the sine divided by the cosine. It is the slope of the radius at angle $\theta$ : rise $\sin\theta$ over run $\cos\theta$ , as in lesson 1.2.1. From the exact table, $\tan 45° = 1$ , $\tan 60° = \sqrt{3} \approx 1.732051$ and $\tan 30° = \frac{1}{\sqrt{3}} \approx 0.57735$ .

Tangent breaks where the cosine is 0, at 90°, 270° and every half turn from them. There the radius is vertical and has no slope. Near $\frac{\pi}{2} \approx 1.570796$ it shoots off: it is 14.1014 at 1.5 and 1255.7656 at 1.57, then $-34.2325$ just past, at 1.6.

A half turn flips the signs of both sine and cosine, and leaves their ratio alone. So tangent repeats every $\pi$ , not every $2\pi$ : $\tan 225° = \tan 45° = 1$ .

Rule: Tangent

\begin{gathered} \tan\theta = \frac{\sin\theta}{\cos\theta} \\ \text{only where } \cos\theta \neq 0 \end{gathered}

The slope of the radius at angle $\theta$ . Its period is $\pi$ .

Sine and cosine make the same wave, a quarter turn apart, repeating every $2\pi$ . Tangent repeats every $\pi$ and shoots off to $\pm$ infinity at each dashed line, where $\cos\theta = 0$ .

\frac{\pi}{2}

import math for degrees in [30, 390, 750, -330]: print(degrees, round(math.sin(math.radians(degrees)), 6)) for t in [1.5, 1.57, 1.6]: print(t, round(math.cos(t), 6), round(math.tan(t), 4)) print(math.tan(math.pi / 2))

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Lesson 1.7.3 writes the point $(\cos\theta, \sin\theta)$ as one complex number, and lesson 2.1.1 needs radians for the limit of $\frac{\sin x}{x}$ at 0.

Common mistakes

Python works in radians. math.sin(90) is the sine of 90 radians, 0.893997, not of 90°. Convert first with math.radians.
Convert the right way round. Degrees to radians multiplies by $\frac{\pi}{180}$ ; radians to degrees multiplies by $\frac{180}{\pi}$ .
Cosine is across, sine is up. The point at 30° is $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ , not the other way round.
Take the sign from the quadrant. $\cos 150°$ is $-\frac{\sqrt{3}}{2}$ : the size of $\cos 30°$ , with a minus sign.
The identity gives two answers. $\sin^2\theta = 0.64$ allows 0.8 and $-0.8$ . The quadrant decides which.
$\sin^2\theta$ means $(\sin\theta)^2$ , not $\sin(\theta^2)$ . At $\frac{\pi}{6}$ they are 0.25 and 0.270734.
Tangent has gaps. At 90° and 270° the cosine is 0 and $\tan\theta$ has no value, whatever Python prints.

Angles, sine, cosine and tangent

0 of 12 correct

1.
Convert 72° to radians (type pi for $\pi$ )
2.
Convert $\frac{5\pi}{6}$ radians to degrees
3.
On a circle of radius 2, how long is the arc cut off by an angle of 1.5 radians?
4.
Give $\sin 60°$ exactly (type sqrt for a square root)
5.
Give $\cos 45°$ exactly
6.
Give $\cos 120°$ exactly
7.
Give $\sin 315°$ exactly
8.
Give $\cos(-60°)$ exactly
9.
$\theta$ is in quadrant II and $\sin\theta = 0.6$ . What is $\cos\theta$ ?
10.
Give $\sin 405°$ exactly
11.
Give $\tan 120°$ exactly
12.
At which angle between 0° and 180° does $\tan\theta$ have no value? (in degrees)

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

2\pi

Angles, the unit circle, sine and cosine

Degrees and radians

Rule: Converting angles

Sine and cosine on the unit circle

Rule: Sine and cosine

Exact values at 30°, 45° and 60°

Signs, and every other angle

Rule: Any angle

The Pythagorean identity

Rule: Pythagorean identity

Periodic functions and their graphs

Rule: Period of sine and cosine

Tangent

Rule: Tangent

Common mistakes

Angles, sine, cosine and tangent

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