Reference angles and signs in each quadrant

About this lesson

Precalculus › Trigonometric functions

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Reference angles and signs in each quadrant

Warm-up

Three quick questions from earlier lessons.

Discover

The point at 150∘150^\circ is the mirror image of the point at 30∘30^\circ across the yy-axis. What is cos⁡150∘\cos 150^\circ?

  1. −32-\dfrac{\sqrt{3}}{2}
  2. 32\dfrac{\sqrt{3}}{2}

Key idea

Every angle has a reference angle: the acute angle between its side and the xx-axis.

sin⁡\sin and cos⁡\cos of an angle have the same size as for its reference angle; the quadrant gives the sign. Quadrant I: both positive. II: sin⁡>0\sin > 0, cos⁡<0\cos < 0. III: both negative. IV: sin⁡<0\sin < 0, cos⁡>0\cos > 0.

Worked example

Find sin⁡225∘\sin 225^\circ.

Quadrant III
225° is between 180° and 270°.
Reference angle: 225∘−180∘=45∘225^\circ - 180^\circ = 45^\circ
Measure from 180°.
sin⁡225∘=−22\sin 225^\circ = -\dfrac{\sqrt{2}}{2}
sin 45° = √2/2, and sine is negative in quadrant III.

Your turn, guided

Your turn: cos⁡300∘\cos 300^\circ.

Reference angle: 360∘−300∘=60∘360^\circ - 300^\circ = 60^\circ
300° is between 270° and 360°: quadrant IV.
cos⁡300∘=12\cos 300^\circ = \dfrac{1}{2}
cos 60° = 1/2, and cosine is positive in quadrant IV.

Practice ★

  1. What is the reference angle of 200∘200^\circ, in degrees?
  2. What is the reference angle of 240∘240^\circ, in degrees?
  3. What is the reference angle of 250∘250^\circ, in degrees?

Watch out!

Practice ★★

  1. Find the exact value of cos⁡(5π4)\cos\left(\frac{5 \pi}{4}\right).
  2. Find the exact value of cos⁡(4π3)\cos\left(\frac{4 \pi}{3}\right).
  3. Find the exact value of cos⁡(2π3)\cos\left(\frac{2 \pi}{3}\right).
  4. Find the exact value of cos⁡(7π4)\cos\left(\frac{7 \pi}{4}\right).
  1. In which quadrant is sin⁡θ<0\sin\theta < 0 and cos⁡θ>0\cos\theta > 0?
    1. Quadrant II
    2. Quadrant III
    3. Quadrant IV

Exit ticket

Three questions to check you have got it.

  1. Find the exact value of cos⁡(5π4)\cos\left(\frac{5 \pi}{4}\right).
  2. Find the exact value of cos⁡(4π3)\cos\left(\frac{4 \pi}{3}\right).
  3. Find the exact value of cos⁡(2π3)\cos\left(\frac{2 \pi}{3}\right).

Summary card

Reference angles

  1. Find the quadrant.
  2. Reference angle: acute angle to the xx-axis (180∘−θ180^\circ - \theta, θ−180∘\theta - 180^\circ or 360∘−θ360^\circ - \theta).
  3. Value of the reference angle, with the quadrant's sign.

Watch out: cosine is negative on the left (II, III), sine below (III, IV).

End of the lesson

Next lesson: Tangent and the reciprocal trigonometric functions Back to the chapter

Answers: Reference angles and signs in each quadrant

A

  1. 2020
  2. 6060
  3. 7070

B

  1. −22- \frac{\sqrt{2}}{2}
  2. −12- \frac{1}{2}
  3. −12- \frac{1}{2}
  4. 22\frac{\sqrt{2}}{2}

C

  1. Quadrant IV

D

  1. −22- \frac{\sqrt{2}}{2}
  2. −12- \frac{1}{2}
  3. −12- \frac{1}{2}

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