Cogito
Trigonometry · Chapter 2 · Lesson 2
Reference Angles and the Four Quadrants
One acute angle can answer for all four.
12 problems · about 21 minutes · F-TF.A.2, F-TF.A.3
What this lesson teaches
The student uses reference angles and quadrant signs to evaluate trigonometric functions of any angle.
- The reference angle is the acute angle to the x-axis.
- It sets the size of the ratio; the quadrant sets the sign.
- Sine follows y, cosine follows x, tangent is positive when they agree.
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is the reference angle for 200°, in degrees?
Answer 20
Why 20°.
What is sin 330° as a decimal?
Answer -0.5
Why −0.5.
Reference angle for 150°, in degrees?
Answer 30
Why 180 − 150.
Reference angle for 300°, in degrees?
Answer 60
Why 360 − 300.
Reference angle for 225°, in degrees?
Answer 45
Why 225 − 180.
Build It Up
The same ideas with more to keep track of.
3 problemssin 150° as a decimal?
Answer 0.5
Why Reference 30°, and sine is positive in quadrant II.
sin 210° as a decimal?
Answer -0.5
Why Reference 30°, and sine is negative in quadrant III.
In which quadrants is tangent positive?
Answer Quadrant I; Quadrant III
Why Tangent is y ÷ x, positive when the two signs agree.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut these angles in order by size of reference angle, smallest first.
Answer 1. 170° 2. 200° 3. 310° 4. 225°
Why The four reference angles are 10°, 20°, 50° and 45°.
cos 300° as a decimal?
Answer 0.5
Why Reference 60°, and cosine is positive in quadrant IV.
The Reference: What is the reference angle for 240°, in degrees?
Answer 60 degrees
Why 60°.
The Sign: In which quadrant are sine and cosine both negative?
Answer Quadrant III.
Why Quadrant III, where x and y are both negative.