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Radians & the unit circle

lesson · about 3 minutes

Radians, exact values, and signs in each quadrant.

Pen and paper is fine · no calculator needed why?

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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

watch out for

practice

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Degrees to radians

worked example

120° is kπ radians. What is k? Give a fraction or a whole number.

Answer: 2/3

  1. Multiply by π/180: 120/180 = 2/3, so 120° = 2π/3 radians.

Radians to degrees

worked example

Convert π radians to degrees.

Answer: 180

  1. π radians is 180°.

Signs by quadrant

worked example

An angle θ has cos θ < 0 and sin θ > 0. Which quadrant is θ in?

  1. quadrant IV
  2. quadrant III
  3. quadrant II
  4. quadrant I

Answer: quadrant II

  1. cos θ < 0 in quadrants II and III.
  2. sin θ > 0 in quadrants I and II.
  3. Both together: quadrant II.

Exact values

worked example

What is the exact value of sin 315°?

  1. −√3/2
  2. −√2/2
  3. √2/2
  4. √3/2

Answer: −√2/2

  1. 315° is 45° short of 360°, in quadrant IV, where sine is negative: sin 315° = −sin 45° = −√2/2.

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