The unit circle

The unit circle is a circle with a radius of 1, centred (centered in the US) on the origin of a graph. Draw an angle θ from the positive x-axis, turning anticlockwise (counterclockwise in the US), and the point where it meets the circle has coordinates (cos θ, sin θ). That one fact gives you sine, cosine and tangent for any angle, including angles bigger than 90° and negative angles.

What is the unit circle?

Every point (x, y) on the circle is exactly 1 unit from the origin. By Pythagoras' theorem, that gives the equation of the unit circle:

Now pick a point on the circle at angle θ and drop a line straight down to the x-axis. You get a right-angled triangle (right triangle in the US) with a hypotenuse of 1, a horizontal side of x and a vertical side of y. SOH CAH TOA says:

Because the hypotenuse is 1, there is nothing to divide by. The x-coordinate is the cosine and the y-coordinate is the sine. That is the whole reason for choosing a radius of 1.

The triangle only works for angles between 0° and 90°, but the coordinates keep going. Past 90° the point carries on round the circle, and its coordinates become the definition of cos θ and sin θ for that angle. That is how trigonometry extends from triangles to every real number, which the US Common Core standards ask students to explain.

Tangent on the unit circle

Tangent is opposite over adjacent, so on the unit circle it is y divided by x:

It is also the gradient (slope) of the line from the origin to the point. At 90° and 270° the point is straight up or down, x = 0, and you can't divide by zero, so tan 90° and tan 270° are undefined.

Unit circle chart in degrees and radians

These are the 16 angles that come up most, with their exact values. The cos column is the x-coordinate of the point and the sin column is the y-coordinate.

As decimals, √3/2 ≈ 0.866, √2/2 ≈ 0.707 and √3/3 ≈ 0.577. You will sometimes see √3/3 written as 1/√3 and √2/2 as 1/√2: they are the same numbers.

Try one. Read the sine and cosine of the marked angle off the circle, then press Mark to check your answer in the course.

The unit circle in radians

A radian is the angle that cuts off an arc of length 1 on the unit circle. The whole circumference is 2π × 1 = 2π, so a full turn is 2π radians, a half turn is π and a quarter turn is π/2. On the unit circle, the angle in radians is the same number as the distance you have walked round the edge.

To convert, use 180° = π radians:

So 150° is 150 × π/180 = 5π/6, and 7π/4 is 7 × 180 ÷ 4 = 315°. A quick check: π/6 is 30°, and every radian angle on the chart is a multiple of π/6 or π/4.

How to memorise the unit circle

Don't learn all 16 rows. Learn the first quarter, 0° to 90°, and get everything else by symmetry. For the first quarter there is a pattern:

Sine counts up √0, √1, √2, √3, √4, all over 2. Cosine is the same list backwards. If you prefer something physical, try the hand trick: hold up your left hand, palm towards you, and give the fingers the angles 0°, 30°, 45°, 60° and 90° from little finger to thumb. Fold down the finger for your angle. The square root of the number of fingers on the little-finger side, over 2, is the sine. The square root of the fingers on the thumb side, over 2, is the cosine. For 30°, one finger is on one side and three on the other, so sin 30° = √1/2 and cos 30° = √3/2.

Two more facts finish the job. At 45° the point sits on the line y = x, so sine and cosine are equal. And 30° and 60° swap their values, because a 30° triangle is a 60° triangle turned on its side.

Unit circle quadrants and signs

The axes cut the circle into four quadrants, numbered anticlockwise from the top right. The sign of each value comes straight from the signs of x and y there.

US students remember this as All Students Take Calculus, going anticlockwise from quadrant I. UK students usually draw it as a CAST diagram, going anticlockwise from quadrant IV. Tangent is positive in quadrant III because a negative divided by a negative is positive.

Reference angles: finding any value

The reference angle is the acute angle between the arm of your angle and the x-axis. Every angle on the circle has the same values as its reference angle, apart from the sign. So any question comes down to three steps:

1. Find the quadrant.
2. Find the reference angle. In quadrant II it is 180° − θ, in quadrant III θ − 180°, and in quadrant IV 360° − θ.
3. Take the first-quadrant value and give it the quadrant's sign.

Worked examples

Example 1: cos 120°

120° is in quadrant II. The reference angle is 180° − 120° = 60°, and cos 60° = 1/2. Cosine is negative in quadrant II, so cos 120° = −1/2.

Example 2: sin(7π/6)

7π/6 is 7 × 30° = 210°, which is in quadrant III. The reference angle is 210° − 180° = 30°, and sin 30° = 1/2. Sine is negative in quadrant III, so sin(7π/6) = −1/2.

Example 3: tan 240°

240° is in quadrant III, with reference angle 240° − 180° = 60°. tan 60° = √3, and tangent is positive in quadrant III, so tan 240° = √3. Check with coordinates: the point is (−1/2, −√3/2), and dividing y by x gives √3, because the two minus signs cancel.

Example 4: solving sin θ = −√3/2

Find every θ with 0 ≤ θ < 2π. Ignore the sign first: sin θ = √3/2 gives a reference angle of π/3. Sine is negative in quadrants III and IV, so there are two answers:

In degrees, those are 240° and 300°. Both points have a y-coordinate of −√3/2 on the chart.

Example 5: using the Pythagorean identity

Every point on the circle satisfies x² + y² = 1. Swap in cos θ and sin θ and you get the Pythagorean identity:

Say cos θ = −5/13 and θ is in quadrant III. Then sin²θ = 1 − 25/169 = 144/169, so sin θ = ±12/13. Sine is negative in quadrant III, so sin θ = −12/13, and tan θ = (−12/13) ÷ (−5/13) = 12/5.

Negative angles and angles over 360°

A negative angle turns clockwise instead of anticlockwise. −60° lands on the same point as 300°, so cos(−60°) = 1/2 and sin(−60°) = −√3/2. Reflecting in the x-axis keeps x and flips y, which gives two rules:

An angle over 360° goes round more than once. Take away 360° (or 2π) until you land between 0° and 360°. For example, 750° − 720° = 30°, so sin 750° = sin 30° = 1/2. Angles that land on the same point, like 30°, 390° and −330°, are called coterminal, and they have the same sin, cos and tan.

From the unit circle to the sine and cosine graphs

Walk round the circle and plot the height of the point against the distance travelled. That is the sine graph. Plot the horizontal position instead and you get the cosine graph.

Here x is the angle in radians and y is sin x. The curve tops out at 1 at π/2, where the point is at the top of the circle, and bottoms out at −1 at 3π/2. After 2π it repeats, because you are back where you started.

Again x is the angle in radians, and y is cos x. Cosine starts at 1, because the point starts at (1, 0). It is the sine graph shifted π/2 to the left. The Trigonometry course builds these waves up from the circle and shows how to stretch and shift them.

Interactive unit circle

Seeing the point move makes the signs and symmetries obvious. Open the trigonometry calculator in unit circle mode and drag the point round: it shows the angle, sin, cos and tan as you go. The unit circle section of the Trigonometry course has an explorer that draws the triangle inside the circle for every angle.

Common mistakes

• Swapping sin and cos. Cosine is x and sine is y, in alphabetical order: (cos θ, sin θ).
• Measuring the angle from the wrong place. Angles start on the positive x-axis and turn anticlockwise. The reference angle is measured to the x-axis, never the y-axis.
• Forgetting the sign. The first-quadrant value is always positive. Decide the sign from the quadrant every time.
• Calling tan 90° infinity. It is undefined, because cos 90° = 0.
• Calculator in the wrong mode. sin 30 is 0.5 in degree mode but about −0.988 in radian mode. Check before an exam.
• Giving one answer to an equation. sin θ = 1/2 has two answers between 0° and 360°: 30° and 150°.

To practise under time pressure, try today's AP Precalculus short timed paper. In the UK, the same daily practice opens as A-level Maths: Algebra & Functions.

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