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PreCalc 2 · Trigonometric Functions

Reference Angles

A reference angle is the acute angle to the x-axis. Find it, then attach the correct sign for the quadrant.

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Key Ideas Easy Medium Hard
What to Remember

How to find it

Quadrant I: the angle itself.

Quadrant II: 180° − θ (or π − θ).

Quadrant III: θ − 180° (or θ − π).

Quadrant IV: 360° − θ (or 2π − θ).

Then use the sign

All Students Take Calculus: Quadrant I all +, II sine +, III tangent +, IV cosine +.

Example: 150° has reference 30°. sin 150° = sin 30° = 1/2.

Level 1

Easy

Example

Find the reference angle for 150°.

Quadrant II: 180° − 150° = 30°.

Answer: 30°

1

Find the reference angle for 120°.

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180° − 120° = 60°.

Answer: 60°

2

Find the reference angle for 210°.

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210° − 180° = 30°.

Answer: 30°

3

Find the reference angle for 315°.

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360° − 315° = 45°.

Answer: 45°

4

Find the reference angle for 330°.

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360° − 330° = 30°.

Answer: 30°

5

Find the reference angle for 240°.

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240° − 180° = 60°.

Answer: 60°

6

Find the reference angle for 225°.

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225° − 180° = 45°.

Answer: 45°

7

Find the reference angle for 19π/6.

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19π/6 − 2π = 19π/6 − 12π/6 = 7π/6.

7π/6 is Quadrant III: 7π/6 − π = π/6.

Answer: π6

Level 2

Medium

Example

Use a reference angle to find cos 150°.

Reference 30°. Quadrant II: cosine is negative.

cos 150° = − cos 30° = −√3/2.

Answer: −√3/2

1

Find sin 210°.

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Reference 30°. Quadrant III: sine is negative.

sin 210° = −1/2.

Answer: −1/2

2

Find tan 315°.

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Reference 45°. Quadrant IV: tangent is negative.

tan 315° = −1.

Answer: −1

3

Find cos 240°.

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Reference 60°. Quadrant III: cosine is negative.

cos 240° = −1/2.

Answer: −1/2

4

Find sin 300°.

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Reference 60°. Quadrant IV: sine is negative.

sin 300° = −√3/2.

Answer: −√3/2

5

Find tan 120°.

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Reference 60°. Quadrant II: tangent is negative.

tan 120° = −√3.

Answer: −√3

6

Find tan 210°.

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Reference 30°. Quadrant III: tangent is positive.

tan 210° = tan 30° = √3/3.

Answer: √3/3

7

Find cos 225°.

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Reference 45°. Quadrant III: cosine is negative.

cos 225° = −√2/2.

Answer: −√2/2

8

Find sin 2π/3.

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Reference π/3. Quadrant II: sine is positive.

sin 2π/3 = √3/2.

Answer: √3/2

Level 3

Hard

Example

Find sin 5π/6.

Reference π/6. Quadrant II: sine is positive.

sin = 1/2.

Answer: 1/2

1

Find cos 7π/6.

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Reference π/6. Quadrant III: cosine is negative.

cos = −√3/2.

Answer: −√3/2

2

Find tan 4π/3.

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Reference π/3. Quadrant III: tangent is positive.

tan = √3.

Answer: √3

3

Find sin 11π/6.

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Reference π/6. Quadrant IV: sine is negative.

sin = −1/2.

Answer: −1/2

4

Find cos 5π/4.

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Reference π/4. Quadrant III: cosine is negative.

cos = −√2/2.

Answer: −√2/2

5

Find tan 7π/4.

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Reference π/4. Quadrant IV: tangent is negative.

tan = −1.

Answer: −1

6

Find sin(−11π/6).

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−11π/6 + 2π = π/6.

sin(−11π/6) = sin(π/6) = 1/2.

Answer: 1/2

7

Find sec 495°.

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495° − 360° = 135°.

Reference of 135° is 45°. Quadrant II: cosine is negative.

sec 495° = 1/cos 135° = 1/(−√2/2) = −√2.

Answer: −√2

8

Find csc 7π/6.

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sin 7π/6 = −1/2, so csc = 1/sin = −2.

Answer: −2

9

Find cot 5π/2.

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5π/2 − 2π = π/2.

cot(π/2) = cos/sin = 0/1 = 0.

Answer: 0

10

Find the reference angle for −3π/8.

Show Answer

−3π/8 is in Quadrant IV (clockwise from the +x-axis).

The acute angle to the x-axis is 3π/8.

Answer: 3π8

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