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You’ve uploaded a worksheet titled “Worksheet – Reference Angles”, which asks students to:

> *Express each of the following in terms of a reference angle. Then give the exact value of each function in simplified radical form.*

This is a standard trigonometry exercise involving reference angles and exact values of trig functions for common angles.

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## ✔ What is a Reference Angle?

A reference angle is the smallest positive acute angle (between 0° and 90°, or 0 and π/2 radians) formed between the terminal side of an angle and the x-axis.

- For any angle θ, its reference angle θ’ is:
- Quadrant I: θ’ = θ
- Quadrant II: θ’ = 180° – θ (or π – θ)
- Quadrant III: θ’ = θ – 180° (or θ – π)
- Quadrant IV: θ’ = 360° – θ (or 2π – θ)

Then, you use the reference angle to find the exact value, applying the correct sign based on the quadrant (ASTC: All Students Take Calculus — All positive in QI, Sine in QII, Tangent in QIII, Cosine in QIV).

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## 📝 Let’s solve each problem step-by-step

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1.

a) cos 45°

- Already in QI → reference angle = 45°
- cos 45° = √2 / 2

✔ Answer: √2 / 2

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b) tan 180°

- On negative x-axis → reference angle = 0°
- tan 180° = sin(180°)/cos(180°) = 0 / (-1) = 0

✔ Answer: 0

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c) sin 210°

- 210° is in QIII → reference angle = 210° – 180° = 30°
- sin is negative in QIII → sin 210° = –sin 30° = –1/2

✔ Answer: –1/2

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d) cos 150°

- QII → ref angle = 180° – 150° = 30°
- cos is negative in QII → cos 150° = –cos 30° = –√3 / 2

✔ Answer: –√3 / 2

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2.

a) sin(–45°)

- Negative angle → add 360° → 315° (QIV)
- ref angle = 360° – 315° = 45°
- sin is negative in QIV → sin(–45°) = –sin 45° = –√2 / 2

✔ Answer: –√2 / 2

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b) cos 330°

- QIV → ref angle = 360° – 330° = 30°
- cos is positive in QIV → cos 330° = cos 30° = √3 / 2

✔ Answer: √3 / 2

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c) sin(–225°)

- Add 360° → 135° (QII)
- ref angle = 180° – 135° = 45°
- sin is positive in QII → sin(–225°) = sin 135° = sin 45° = √2 / 2

✔ Answer: √2 / 2

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d) cos 240°

- QIII → ref angle = 240° – 180° = 60°
- cos is negative in QIII → cos 240° = –cos 60° = –1/2

✔ Answer: –1/2

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3.

a) sin 120°

- QII → ref angle = 180° – 120° = 60°
- sin positive → sin 120° = sin 60° = √3 / 2

✔ Answer: √3 / 2

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b) cos(–60°)

- Add 360° → 300° (QIV)
- ref angle = 360° – 300° = 60°
- cos positive in QIV → cos(–60°) = cos 60° = 1/2

✔ Answer: 1/2

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c) sin(–135°)

- Add 360° → 225° (QIII)
- ref angle = 225° – 180° = 45°
- sin negative in QIII → sin(–135°) = –sin 45° = –√2 / 2

✔ Answer: –√2 / 2

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d) tan(–30°)

- Add 360° → 330° (QIV)
- ref angle = 360° – 330° = 30°
- tan negative in QIV → tan(–30°) = –tan 30° = –1/√3 = –√3 / 3

✔ Answer: –√3 / 3

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4.

a) cos 210°

- QIII → ref angle = 210° – 180° = 30°
- cos negative → cos 210° = –cos 30° = –√3 / 2

✔ Answer: –√3 / 2

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b) cos 90°

- On y-axis → reference angle = 90° (but technically, it's a quadrantal angle)
- cos 90° = 0

✔ Answer: 0

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c) sin(–120°)

- Add 360° → 240° (QIII)
- ref angle = 240° – 180° = 60°
- sin negative in QIII → sin(–120°) = –sin 60° = –√3 / 2

✔ Answer: –√3 / 2

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d) sin(–315°)

- Add 360° → 45° (QI)
- ref angle = 45°
- sin positive → sin(–315°) = sin 45° = √2 / 2

✔ Answer: √2 / 2

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5.

a) sin(π/6)

- π/6 = 30° → QI → ref angle = π/6
- sin(π/6) = 1/2

✔ Answer: 1/2

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b) cos(π/3)

- π/3 = 60° → QI
- cos(π/3) = 1/2

✔ Answer: 1/2

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c) tan(2π/3)

- 2π/3 = 120° → QII
- ref angle = π – 2π/3 = π/3
- tan negative in QII → tan(2π/3) = –tan(π/3) = –√3

✔ Answer: –√3

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d) cos(3π/4)

- 3π/4 = 135° → QII
- ref angle = π – 3π/4 = π/4
- cos negative → cos(3π/4) = –cos(π/4) = –√2 / 2

✔ Answer: –√2 / 2

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6.

a) cos(5π/6)

- 5π/6 = 150° → QII
- ref angle = π – 5π/6 = π/6
- cos negative → cos(5π/6) = –cos(π/6) = –√3 / 2

✔ Answer: –√3 / 2

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b) sin(–π/4)

- Add 2π → 7π/4 (QIV)
- ref angle = 2π – 7π/4 = π/4
- sin negative in QIV → sin(–π/4) = –sin(π/4) = –√2 / 2

✔ Answer: –√2 / 2

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c) tan(–π/3)

- Add 2π → 5π/3 (QIV)
- ref angle = 2π – 5π/3 = π/3
- tan negative in QIV → tan(–π/3) = –tan(π/3) = –√3

✔ Answer: –√3

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d) sin(5π/6)

- 5π/6 = 150° → QII
- ref angle = π – 5π/6 = π/6
- sin positive → sin(5π/6) = sin(π/6) = 1/2

✔ Answer: 1/2

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7.

a) cos(2π + π/6)

- 2π + π/6 = 13π/6 → subtract 2π → π/6 (same as 30°)
- cos(π/6) = √3 / 2

✔ Answer: √3 / 2

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b) sin(–π/2)

- Quadrantal angle → sin(–π/2) = –1

✔ Answer: –1

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c) tan(–3π/4)

- Add 2π → 5π/4 (QIII)
- ref angle = 5π/4 – π = π/4
- tan positive in QIII → tan(–3π/4) = tan(5π/4) = tan(π/4) = 1

✔ Answer: 1

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d) cos(3π/2)

- Quadrantal angle → cos(3π/2) = 0

✔ Answer: 0

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## ✍️ Final Answers Summary (for easy copying):

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1.

a) √2 / 2
b) 0
c) –1/2
d) –√3 / 2

2.

a) –√2 / 2
b) √3 / 2
c) √2 / 2
d) –1/2

3.

a) √3 / 2
b) 1/2
c) –√2 / 2
d) –√3 / 3

4.

a) –√3 / 2
b) 0
c) –√3 / 2
d) √2 / 2

5.

a) 1/2
b) 1/2
c) –√3
d) –√2 / 2

6.

a) –√3 / 2
b) –√2 / 2
c) –√3
d) 1/2

7.

a) √3 / 2
b) –1
c) 1
d) 0

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## 🎯 Key Concepts Reinforced:

- Reference angles help reduce any angle to an equivalent acute angle.
- Signs depend on the quadrant (ASTC).
- Exact values come from special triangles (30-60-90, 45-45-90).
- Negative angles are measured clockwise; add 360° (or 2π) to get coterminal positive angle.
- Quadrantal angles (like 0°, 90°, 180°, etc.) have known exact values.

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