Sine and cosine on the unit circle

Picture a Ferris wheel at the fair. Your seat starts on the right, level with the axle, and the wheel turns anticlockwise. You go up to the top, then over to the left, down to the bottom and back. With sine and cosine you can work out where your seat is for any angle: how high it is, and how far left or right. The tool for this is the unit circle.

From triangle to unit circle

You may already know sine, cosine and tangent from right triangles. The memory aid is SOH CAH TOA:

sin α = opposite ÷ hypotenuse
cos α = adjacent ÷ hypotenuse
tan α = opposite ÷ adjacent

Take the triangle with sides 3, 4 and 5. The angle α is opposite the side of length 3. Then:

sin α = 3 ÷ 5 = 0.6
cos α = 4 ÷ 5 = 0.8

Now divide every side by 5. The shape stays the same, so the angle α does too. The sides become 0.6, 0.8 and 1. With a hypotenuse of 1 there is nothing left to divide by. sin α is simply the opposite side, and cos α is the adjacent side.

From triangle to unit circle

every side ÷ 5

Show the numbers

Side	Triangle 3, 4, 5	Divided by 5
opposite	3	0.6
adjacent	4	0.8
hypotenuse	5	1

Put that small triangle on a pair of axes, with the angle α at the origin O. The corner P then lies on a circle of radius 1: the unit circle. For P:

x = cos α
y = sin α

This is the definition for every angle, including angles above 90°. A right triangle no longer fits there, but there is always a point P on the circle.

Turn P around the circle

You measure an angle from the positive x-axis, anticlockwise. Drag P around below, or use the slider. The line along the x-axis is cos α. The line going up or down is sin α.

Turn P around the unit circle

• cos α (x)
• sin α (y)

Angle α: 40°

cos 40° (x)

≈ 0.766

sin 40° (y)

≈ 0.643

tan 40°

≈ 0.839

Quadrant

I: cos +, sin +

Reflect in the y-axis: 180° − αReflect in the x-axis: −α

Show the signs per quadrant

Quadrant	Angles	cos	sin	tan
I	0° < α < 90°	+	+	+
II	90° < α < 180°	−	+	−
III	180° < α < 270°	−	−	+
IV	270° < α < 360°	+	−	−

The axes split the circle into four quadrants. In each quadrant x and y have a fixed sign, so cos α and sin α do too:

• Quadrant I, from 0° to 90°: cos +, sin +
• Quadrant II, from 90° to 180°: cos −, sin +
• Quadrant III, from 180° to 270°: cos −, sin −
• Quadrant IV, from 270° to 360°: cos +, sin −

Example: α = 140°. P is in quadrant II, left of the y-axis and above the x-axis. So x is negative and y is positive:

cos 140° ≈ −0.766
sin 140° ≈ 0.643

On the axes the values come out exactly. At 90° P is at the top, at (0, 1). So cos 90° = 0 and sin 90° = 1. At 180° P is on the left, at (−1, 0): cos 180° = −1 and sin 180° = 0. At 270° P is at the bottom: sin 270° = −1.

Why sin 140° = sin 40°

Reflect P at 40° in the y-axis. The image P′ lands at 180° − 40° = 140°. Both points are at the same height. So their y is the same, and so is their sine. Only the sign of x changes.

sin 140° = sin 40°

sin 40°

≈ 0.643

sin 140°

≈ 0.643

cos 40°

≈ 0.766

cos 140°

≈ −0.766

Show the numbers

Angle	sin α	cos α
40°	0.643	0.766
140°	0.643	−0.766

sin(180° − α) = sin α
cos(180° − α) = −cos α

Example: you know that sin 20° ≈ 0.342. Since 160° = 180° − 20°, you also know sin 160°:

sin 160° = sin 20° ≈ 0.342

cos(−α) = cos α

A negative angle turns clockwise. At −40° you land on the image of P reflected in the x-axis. That point is straight below P: the same x, and y with the opposite sign.

cos(−40°) = cos 40°

cos 40°

≈ 0.766

cos(−40°)

≈ 0.766

sin 40°

≈ 0.643

sin(−40°)

≈ −0.643

Show the numbers

Angle	sin α	cos α
40°	0.643	0.766
−40°	−0.643	0.766

cos(−α) = cos α
sin(−α) = −sin α

So cos(−40°) = cos 40° ≈ 0.766 and sin(−40°) = −sin 40° ≈ −0.643. The angle −40° gives the same point as 360° − 40° = 320°. In the circle you can turn, above, the buttons show both reflections for any angle.

Pythagoras and tangent

Look at the triangle inside the circle again. The two shorter sides are cos α and sin α, and the hypotenuse is 1. Pythagoras gives:

sin²α + cos²α = 1

Here sin²α means (sin α)². Check it with the triangle 0.6, 0.8 and 1:

0.6² + 0.8² = 0.36 + 0.64 = 1

It holds for every angle. P is always on the circle, so x² + y² = 1. Squaring gets rid of any minus sign. At 140°, to three decimal places: 0.413 + 0.587 = 1.000.

Pythagoras and tangent in one picture

cos²α + sin²α = 0.8² + 0.6² = 0.64 + 0.36 = 1
tan α = 0.6 ÷ 0.8 = 0.75

Show the numbers

cos α	sin α	tan α	α
0.8	0.6	0.75	≈ 36.9°

This rule gives you cos α when you know sin α. Example: sin α = 0.6 and α is in quadrant II.

cos²α = 1 − 0.6² = 1 − 0.36 = 0.64
cos α = −√0.64 = −0.8

The square root gives 0.8 or −0.8. In quadrant II the cosine is negative, so cos α = −0.8. The angle itself is about 143.1°.

The tangent comes from the sine and the cosine:

tan α = sin α ÷ cos α

For the 3, 4, 5 triangle that gives 0.6 ÷ 0.8 = 0.75. That is exactly opposite ÷ adjacent = 3 ÷ 4. In the picture, tan α is the height where the line through O and P meets the line x = 1. At 45° sin α and cos α are equal, so tan 45° = 1. At 90° cos α = 0. You cannot divide by 0, so tan 90° is undefined.

The graphs from 0° to 360°

Put the angle on the horizontal axis and the y of P on the vertical axis. That gives the graph of sin α. Do the same with the x of P, and you get cos α. Drag the vertical line, tap the graph or use the slider. P in the circle above turns with it.

The graphs of sin α and cos α

• sin α
• cos α

Angle α: 40°

sin 40°

≈ 0.643

cos 40°

≈ 0.766

Show the angle with the same sine

Show the numbers

Angle α	sin α	cos α
0°	0.000	1.000
30°	0.500	0.866
45°	0.707	0.707
60°	0.866	0.500
90°	1.000	0.000
120°	0.866	−0.500
135°	0.707	−0.707
150°	0.500	−0.866
180°	0.000	−1.000
210°	−0.500	−0.866
225°	−0.707	−0.707
240°	−0.866	−0.500
270°	−1.000	0.000
300°	−0.866	0.500
315°	−0.707	0.707
330°	−0.500	0.866
360°	0.000	1.000

What the graph shows:

• sin α starts at 0, is 1 at 90°, 0 at 180°, −1 at 270° and 0 again at 360°.
• cos α starts at 1, is 0 at 90°, −1 at 180°, 0 at 270° and 1 again at 360°.
• Both waves stay between −1 and 1, because P is on a circle of radius 1.
• The cosine graph is the sine graph shifted 90° to the left: cos α = sin(α + 90°).

The sine graph is the height of your seat on the Ferris wheel, measured from the axle. If the wheel has a radius of 10 m, everything is 10 times as big. After turning 30° you are 10 × sin 30° = 10 × 0.5 = 5 m above the axle.

Solving sin α = 0.5

Find every angle between 0° and 360° with sin α = 0.5. The calculator gives you one, with the inverse sine:

sin⁻¹(0.5) = 30°

There is a second angle. The line y = 0.5 crosses the circle twice. The second point is the reflection in the y-axis:

α = 180° − 30° = 150°

sin α = 0.5 has two solutions

sin 30°

= 0.5

sin 150°

= 0.5

Show the numbers

Angle α	sin α
30°	0.5
150°	0.5

Check: sin 150° = sin 30° = 0.5. The graph shows the same thing. The line y = 0.5 crosses the sine wave at 30° and at 150°.

A negative value works the same way. Solve sin α = −0.5. The calculator gives sin⁻¹(−0.5) = −30°. That angle is the same point as 360° − 30° = 330°. Its reflection in the y-axis is at 180° + 30° = 210°. So α = 210° or α = 330°.

For cosine you reflect in the x-axis. Solve cos α = 0.5. The calculator gives cos⁻¹(0.5) = 60°. The second solution is 360° − 60° = 300°.

Try it yourself