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courses › Trigonometry

Radians & the unit circle

level 32 course

Radians, exact values, and signs in each quadrant.

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Builds on: Right-triangle trig (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

The unit circle has radius 1 and its center at (0, 0). Turn an angle θ counterclockwise from the positive x-axis, and the point where it lands is (cos θ, sin θ). So cosine is the x-coordinate and sine is the y-coordinate.

Angles can also be measured in radians: π radians is a half turn, 180°. The four quarters of the plane are the quadrants I to IV, counterclockwise from the top right. The signs of x and y there give the signs of cosine and sine.

Techniques

Swap π for 180°

Converting between degrees and radians.

  1. Degrees to radians: divide the degrees by 180 and reduce. That fraction is k in kπ.
  2. Radians to degrees: replace π with 180 and multiply out.
worked example

Example: Convert 7π/4 radians to degrees.

  1. Replace π with 180: 7 × 180 ÷ 4.
  2. 1,260 ÷ 4 = 315.

Answer: 315

Signs by quadrant

You want the sign of sin, cos or tan in a quadrant.

  1. Cosine follows x: positive on the right, in quadrants I and IV.
  2. Sine follows y: positive on top, in quadrants I and II.
  3. Tangent is sine ÷ cosine: positive where their signs match, in I and III.
worked example

Example: An angle θ has sin θ < 0 and tan θ > 0. Which quadrant is θ in? Give the quadrant as a number from 1 to 4.

  1. sin θ < 0 below the x-axis: quadrants III and IV.
  2. tan θ > 0 where sine and cosine match: I and III.
  3. Both hold in quadrant III.

Answer: 3

Reference angle, then the sign

The exact sin, cos or tan of a multiple of 30° or 45°.

  1. Find the reference angle, the gap to the x-axis: 180° − θ in II, θ − 180° in III, 360° − θ in IV.
  2. Take the value at the reference angle, as in a right triangle.
  3. Give it the sign of the quadrant.
  4. On an axis, read the point instead: at 180° it is (−1, 0).
worked example

Example: What is the exact value of cos 240°? Give a fraction.

  1. 240° is 60° past 180°, in quadrant III.
  2. cos 60° = 1/2.
  3. Cosine is negative in quadrant III, so the answer is −1/2.

Answer: −1/2

Tips by skill

  • TipDegrees to radians: Divide the degrees by 180 and reduce the fraction. That fraction is k.
  • TipRadians to degrees: π radians is 180°. Replace π with 180 and multiply out.
  • TipSigns by quadrant: Cosine is positive on the right (I, IV), sine on top (I, II), and tangent where they match (I, III).
  • TipExact values: Find the reference angle to the x-axis, take its value, and give it the quadrant's sign. On an axis, read the point's coordinates instead.

Watch out for

  • Using 360° for π. π radians is a half turn, 180°.
  • Dividing the wrong way: 135° is 135/180 of π, not 180/135.
  • Getting the size right and the sign wrong. Find the quadrant before you write the answer.
  • Swapping sine and cosine on the circle. Cosine is the x-coordinate; sine is the y-coordinate.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
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  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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