Math Vale

🏔️ Trigonometry Peaks · Trigonometry

The Unit Circle

Extend sine and cosine to all angles using coordinates on the unit circle, and read exact values off it.

In short

  • On the unit circle the point at angle t is (cos t, sin t): cosine is the x-coordinate, sine the y-coordinate.
  • The exact values for 30°, 45° and 60° come from the 30-60-90 and 45-45-90 triangles.
  • The reference angle (measured to the x-axis) gives the size; the quadrant gives the sign.
  • A trigonometric equation usually has two solutions in each full turn, and a calculator shows only one.

Sine and cosine as coordinates

Right-triangle trigonometry only handles angles between 0° and 90°. The unit circle — the circle of radius 1 centred at the origin — extends the definitions to every angle.

Start on the positive x-axis and turn anticlockwise through an angle t. The point where you land has coordinates

(cos t, sin t)

Cosine is the x-coordinate (across) and sine is the y-coordinate (up). For an angle in the first quadrant this agrees exactly with the old definitions, because the radius is 1 and the hypotenuse is therefore 1.

Two consequences are immediate. Since the point lies on a circle of radius 1, its coordinates satisfy x2 + y2 = 1, which is the identity sin2(t) + cos2(t) = 1. And since no coordinate can exceed the radius, sine and cosine always lie between -1 and 1.

The third ratio is tan t = sin t / cos t, the y-coordinate divided by the x-coordinate — which is the slope of the radius.

The special angles

Three angles have exact values worth memorising, all coming from two familiar triangles.

The 45-45-90 triangle (a square cut along its diagonal) gives

sin 45° = cos 45° = sqrt(2)/2, tan 45° = 1

The 30-60-90 triangle (an equilateral triangle cut in half) gives

sin 30° = 1/2, cos 30° = sqrt(3)/2, tan 30° = sqrt(3)/3 sin 60° = sqrt(3)/2, cos 60° = 1/2, tan 60° = sqrt(3)

The quadrantal angles come straight from the coordinates:

0°: (1, 0) 90°: (0, 1) 180°: (-1, 0) 270°: (0, -1)

so sin 0° = 0, cos 90° = 0, sin 270° = -1, and so on. Notice that tan 90° cannot be worked out at all, because it would require dividing by cos 90° = 0.

Reference angles

Every angle has a reference angle: the acute angle between its arm and the x-axis. It is always measured to the x-axis, never to the y-axis, and it is always between 0° and 90°.

quadrant 2: reference = 180 - t quadrant 3: reference = t - 180 quadrant 4: reference = 360 - t

So 150° has reference angle 30°, 210° has reference angle 30°, and 300° has reference angle 60°.

The reference angle gives the size of every ratio. The point at 150° is the mirror image of the point at 30°, so its coordinates have the same magnitudes: sin 150° = 1/2 and cos 150° = -sqrt(3)/2.

That leaves only the sign to determine, which is what the quadrant is for.

Signs, and solving for an angle

Since cosine is the x-coordinate and sine is the y-coordinate, the signs follow from the quadrant:

quadrant 1: x > 0, y > 0 -> all three ratios positive quadrant 2: x < 0, y > 0 -> only sine positive quadrant 3: x < 0, y < 0 -> only tangent positive (negative over negative) quadrant 4: x > 0, y < 0 -> only cosine positive

The mnemonic All Students Take Calculus names the ratio that stays positive in quadrants 1, 2, 3 and 4.

Working backwards, an equation such as sin t = 1/2 has two solutions in one full turn, not one. The reference angle is 30°, and sine is positive in quadrants 1 and 2, so

t = 30° or t = 180 - 30 = 150°

For cos t = 1/2, cosine is positive in quadrants 1 and 4, giving t = 60° or t = 300°. A calculator returns only one of the two; the unit circle supplies the other.

Worked examples

Example 1

Find the exact value of cos(210°).

  1. 210° is between 180° and 270°, so it is in the third quadrant.
  2. Its reference angle is 210 - 180 = 30°, and cos 30° = sqrt(3)/2.
  3. In the third quadrant the x-coordinate is negative, so the cosine is negative.
  4. cos(210°) = -sqrt(3)/2.

Example 2

Find all angles t with 0° <= t < 360° for which sin t = sqrt(2)/2.

  1. The reference angle is 45°, since sin 45° = sqrt(2)/2.
  2. The value is positive, and sine is positive in quadrants 1 and 2.
  3. Quadrant 1 gives t = 45°; quadrant 2 gives t = 180 - 45 = 135°.
  4. The solutions are 45° and 135°.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Find the exact value of sin(60°). Write square roots as sqrt(...), for example sqrt(3)/2 or -1/2. Do not give a decimal.

Answer: sqrt(3)/2

  1. 60° has reference angle 60°.
  2. On the unit circle the point is (cos, sin) = (1/2, sqrt(3)/2).
  3. sin(60°) = sqrt(3)/2

Problem 2

Difficulty 3 of 5

Find the exact value of tan(210°). Write square roots as sqrt(...), for example sqrt(3) or -sqrt(3)/3. Do not give a decimal.

Answer: sqrt(3)/3

  1. tan(210°) = sin(210°) / cos(210°)
  2. = (-1/2) / (-sqrt(3)/2)
  3. = sqrt(3)/3

Problem 3

Difficulty 4 of 5

What is the reference angle for 300°?

Answer: 60 degrees

  1. 300° lies in quadrant 4.
  2. Reference angle = 360 - 300
  3. Reference angle = 60°

Common mistakes

  • Swapping sine and cosine — cosine is across, sine is up.
  • Measuring the reference angle to the y-axis instead of the x-axis.
  • Getting the size right but the sign wrong by ignoring the quadrant.
  • Giving only the calculator answer when a second solution exists in the same turn.

What you should be able to do

  • Give exact sine and cosine for the special angles.
  • Use reference angles and quadrant signs.
  • Relate coordinates on the unit circle to (cos t, sin t).
  • Evaluate trig functions of angles beyond one full turn.

Where this fits in the curriculum

Common Core

  • HSF-TF.A.2

    High school — Explain how the unit circle extends sine and cosine to all real numbers, traced counterclockwise.

  • HSF-TF.A.3

    High school — Use special triangles to find sine, cosine and tangent for π/3, π/4 and π/6, and use the unit circle for π − x, π + x and 2π − x.

    HSF-TF.A.3 is a (+) standard — Common Core marks it as beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II.

SAT

  • Additional Topics in Math

    Trigonometric values of angles beyond the right triangle.

    The SAT reaches the unit circle only lightly — enough for radian measure and the sine/cosine of a standard angle, not the full precalculus treatment.

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