Math 130 — Quiz 10 Notes

Section 5.3
Reference Angles & Trig Functions
Fall 2026 planned coverage
Canvas date: Nov 11, 2:10 PM
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Reference Angle Formulas  (θ′)
Q I θ′ = θ
Q II θ′ = 180° − θ
Q III θ′ = θ − 180°
Q IV θ′ = 360° − θ

For radians, replace 180° with π and 360° with 2π.

Negative angle? Add 360° (or 2π) first to get a co-terminal angle, then identify quadrant.

The 3-Step Process — use on every trig value problem
1
Find θ′
Identify the quadrant, then apply the reference-angle formula.
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2
Evaluate at θ′
Use a special triangle to get the magnitude — always positive.
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3
Apply the Sign
Use the ASTC chart to assign +/− based on the original quadrant.
ASTC Sign Chart
S Q II
sin, csc ✓
cos & tan −
A Q I
ALL positive
T Q III
tan, cot ✓
sin & cos −
C Q IV
cos, sec ✓
sin & tan −

"All Students Take Calculus"

Special Triangle Values
30 – 60 – 90
√3 1 2 30° 60°
sin 30° = ½
cos 30° = √3/2
sin 60° = √3/2
cos 60° = ½
tan 30° = √3/3
tan 60° = √3
45 – 45 – 90
1 1 √2 45° 45°
sin 45° = √2/2
cos 45° = √2/2
tan 45° = 1
Six Trig Functions from a Point (x, y)

First compute: r = √(x² + y²)  (always > 0)

sin θ= y/r
csc θ= r/y
cos θ= x/r
sec θ= r/x
tan θ= y/x
cot θ= x/y

Signs come from x and y directly — no need for ASTC. Verify with ASTC as a sanity-check.

Finding Two Angles in [0°, 360°]

Example: sin θ = −√3/2

Step 1 Ignore sign → reference angle θ′ = 60° Step 2 Negative sine → Q III & Q IV only Step 3 Build angles:
Q III: 180° + 60° = 240°
Q IV: 360° − 60° = 300°

Negative-value quadrant summary:

sin < 0 → Q III, IV sin > 0 → Q I, II cos < 0 → Q II, III cos > 0 → Q I, IV tan < 0 → Q II, IV tan > 0 → Q I, III
Solving a Right Triangle
  • Missing side: Pythagorean Theorem — a² + b² = c²
  • Find angle from two sides: use inverse trig
      sin⁻¹(opp/hyp),  cos⁻¹(adj/hyp),  tan⁻¹(opp/adj)
  • Third angle: angles sum to 180° → α = 90° − β
  • Check: α + β + 90° = 180° ✓

Example (Quiz Q10): legs 21 & b, hyp 29
b = √(29²−21²) = √400 = 20
β = sin⁻¹(20/29) ≈ 43.6°  ·  α ≈ 46.4°

Common Errors to Avoid
  • Forgetting to negate after finding the reference-angle value (sign step).
  • Using the reference angle itself as the final answer — it must be acute.
  • Working with a negative angle directly instead of converting to a co-terminal first.
  • Writing cos = 4/5 when x = −4 (ignoring the negative x).
  • Mixing up sin/cos in Step 2 (e.g., using sin 60° when you need cos 60°).
  • For reciprocal functions: csc/sec share the sign of sin/cos — a positive csc means positive sin.
  • Confusing θ′ formula: Q III uses θ − 180°, not 180° − θ.
Worked Example — Evaluating cos(240°) via the 3-Step Process
STEP 1
Find θ′

240° is between 180°–270° → Q III
θ′ = 240° − 180° = 60°

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STEP 2
Evaluate at θ′

From the 30-60-90 triangle:
cos(60°) = 1/2  ← magnitude only

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STEP 3
Apply the Sign

ASTC: Q III → only tangent is positive.
Cosine is negative → cos(240°) = −1/2 ✓

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POINT EXAMPLE
From (−4, 3)

r = √(16+9) = 5
sin θ = 3/5  ·  cos θ = −4/5
tan θ = −3/4  (Q II ✓ via ASTC)