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Unit circle

Every standard angle, in degrees and radians, with exact values.

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  • Arithmetic to calculus

The unit circle is a circle of radius 1 centred on the origin, and it is the fastest way to read a trig value exactly rather than as a decimal. Every point on it is (cos θ, sin θ), so the x coordinate is the cosine and the y coordinate is the sine. Tap any angle above to see its exact values. The solver below handles the equations that use them.

Every angle at a glance

All sixteen standard angles with their exact values. The point on the circle is (cos θ, sin θ) — cosine first.

The unit circle: degrees, radians and the exact cosine, sine and tangent of every standard angle
AngleRadianscossintan
0°0100
30°π/6√3/21/2√3/3
45°π/4√2/2√2/21
60°π/31/2√3/2√3
90°π/201undefined
120°2π/3−1/2√3/2−√3
135°3π/4−√2/2√2/2−1
150°5π/6−√3/21/2−√3/3
180°π−100
210°7π/6−√3/2−1/2√3/3
225°5π/4−√2/2−√2/21
240°4π/3−1/2−√3/2√3
270°3π/20−1undefined
300°5π/31/2−√3/2−√3
315°7π/4√2/2−√2/2−1
330°11π/6√3/2−1/2−√3/3

How to read it

Four facts do almost all the work. Learn these and the other twelve angles follow.

  1. x is cosine, y is sine

    The point at angle θ is (cos θ, sin θ), always in that order. Sine is the height, cosine is how far across.

  2. Only three values appear

    Every first-quadrant value is 1/2, √2/2 or √3/2, in that order as the angle grows from 30° to 45° to 60°. Everything else is one of these with a sign.

  3. The sign comes from the quadrant

    All positive in the first, sine only in the second, tangent only in the third, cosine only in the fourth. Quadrant two is above and left, so y is positive and x is negative.

  4. tan is sine over cosine

    tan θ = sin θ / cos θ, so it is undefined wherever the cosine is zero — at 90° and 270°, where the circle is at its top and bottom.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

What is the exact value of cos(2π/3)?

cos⁡2π3\cos\tfrac{2\pi}{3}
  1. Convert to degrees
    2π3=120∘\tfrac{2\pi}{3} = 120^\circ
  2. Find the reference angle
    180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ
  3. Apply the quadrant sign
    quadrant II: cosine is negative\text{quadrant II: cosine is negative}
  4. Read the value
    cos⁡120∘=−12\cos 120^\circ = -\tfrac{1}{2}
Answer
−12-\tfrac{1}{2}

Solve sin(x) = 1/2 for 0 ≤ x < 2π

sin⁡x=12\sin x = \tfrac{1}{2}
  1. Find where sine is 1/2
    x=π6 (30∘)x = \tfrac{\pi}{6}\ (30^\circ)
  2. Sine is also positive in quadrant II
    x=π−π6=5π6x = \pi - \tfrac{\pi}{6} = \tfrac{5\pi}{6}
Answer
x=π6, 5π6x = \tfrac{\pi}{6},\ \tfrac{5\pi}{6}

A trig equation almost always has two answers in one turn, one per quadrant where the sign fits.

Convert 3π/4 radians to degrees

3π4 rad\tfrac{3\pi}{4}\ \text{rad}
  1. Multiply by 180 over pi
    3π4×180π\tfrac{3\pi}{4} \times \tfrac{180}{\pi}
  2. The pi cancels
    3×1804=135∘\tfrac{3 \times 180}{4} = 135^\circ
Answer
135∘135^\circ

Solve tan(x) = 1 for 0 ≤ x < 2π

tan⁡x=1\tan x = 1
  1. Tangent is 1 where sine equals cosine
    x=π4 (45∘)x = \tfrac{\pi}{4}\ (45^\circ)
  2. Tangent repeats every half turn
    x=π4+π=5π4x = \tfrac{\pi}{4} + \pi = \tfrac{5\pi}{4}
Answer
x=π4, 5π4x = \tfrac{\pi}{4},\ \tfrac{5\pi}{4}

Tangent has a period of π, not 2π, so its solutions are half a turn apart rather than mirrored.

Where marks get lost

Reading the coordinates the wrong way round

The point is (cos θ, sin θ). Sine is the second number, because sine is the height. Swapping them turns every 30° answer into a 60° one.

Forgetting the second solution

sin x = 1/2 has two answers in one turn, not one. Find the reference angle, then ask which quadrants carry that sign.

Working in the wrong angle unit

sin(30) is 0.5 in degrees and −0.988 in radians. Calculus is in radians; most geometry is in degrees.

Questions

What is the unit circle?

A circle of radius 1 centred on the origin. Every point on it is (cos θ, sin θ) for the angle θ measured anticlockwise from the positive x axis, which turns trigonometry into reading coordinates.

How do I memorise the unit circle?

Do not memorise 16 points. Learn the first quadrant — 1/2, √2/2, √3/2 as the angle grows — then get every other point by reflecting and applying the quadrant sign.

How do I convert radians to degrees?

Multiply by 180/π. So 3π/4 becomes 3 × 180 ÷ 4 = 135°. Going the other way, multiply degrees by π/180.

Why is tan undefined at 90 degrees?

tan θ = sin θ / cos θ, and cos 90° = 0. Dividing by zero has no value, so the tangent has a vertical asymptote there rather than a number.

What does the mnemonic All Students Take Calculus mean?

It names which function is positive in each quadrant, going anticlockwise: All in the first, Sine in the second, Tangent in the third, Cosine in the fourth.

What is a reference angle?

The acute angle between the terminal side and the x axis. It gives the size of the value; the quadrant gives the sign. For 210° the reference angle is 30°, so sin 210° = −1/2.

Why are the values exact rather than decimals?

Because √3/2 is the answer and 0.8660254 is a rounding of it. Exact forms stay exact through later steps, which matters once the value goes into an integral or an identity.

Does the circle work for angles over 360 degrees?

Yes. Subtract full turns until the angle lands between 0° and 360°, and read that point. 750° is 750 − 720 = 30°, so it has the same values.