Trigonometry

Trigonometry makes up about 7-10% of the ACT Math section (4-6 questions). It tests your knowledge of trigonometric ratios, the unit circle, identities, and the laws of sines and cosines. Mastering these core concepts can give you a significant edge on test day.

SOH-CAH-TOA (Right Triangle Trig)

For a right triangle with an acute angle θ:

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent

The reciprocal functions are: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.

Example 1: A right triangle has legs of 5 and 12. Find sin θ, cos θ, and tan θ for the angle opposite the side of length 5.
Hypotenuse = √(25 + 144) = √169 = 13 (5-12-13 triple).
sin θ = 5/13, cos θ = 12/13, tan θ = 5/12.

Example 2: From a point 50 feet from the base of a building, the angle of elevation to the roof is 60°. How tall is the building?
tan 60° = height/50. Since tan 60° = √3, height = 50√3 ≈ 86.6 feet.

Example 3: If sin θ = 7/25, and θ is acute, find cos θ.
cos²θ = 1 − sin²θ = 1 − 49/625 = 576/625. cos θ = 24/25 (positive since θ is acute). So cos θ = 24/25.

The Unit Circle

The unit circle has radius 1 centered at the origin. For any angle θ, the point on the circle is (cos θ, sin θ).

Key angles to memorize:

• 0° → sin = 0, cos = 1
• 30° (π/6) → sin = 1/2, cos = √3/2
• 45° (π/4) → sin = √2/2, cos = √2/2
• 60° (π/3) → sin = √3/2, cos = 1/2
• 90° (π/2) → sin = 1, cos = 0

Quadrant sign rule — "All Students Take Calculus":
QI: All positive | QII: Sine positive | QIII: Tangent positive | QIV: Cosine positive

Example 1: Find sin(150°).
150° is in Quadrant II (sine is positive). Reference angle = 180° − 150° = 30°.
sin(150°) = sin(30°) = 1/2.

Example 2: Find cos(225°).
225° is in Quadrant III (cosine is negative). Reference angle = 225° − 180° = 45°.
cos(225°) = −cos(45°) = −√2/2.

Example 3: Find tan(330°).
330° is in Quadrant IV (tangent is negative). Reference angle = 360° − 330° = 30°.
tan(330°) = −tan(30°) = −√3/3.

Radian and Degree Conversion

Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π.
Example 1: 120° = 120 × π/180 = 2π/3 radians.
Example 2: 5π/4 radians = (5π/4) × (180/π) = 225°.

Fundamental Trig Identities

• sin²θ + cos²θ = 1
• tan θ = sin θ / cos θ
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Example: Simplify (1 − cos²θ)/sin θ.
Since 1 − cos²θ = sin²θ, the expression becomes sin²θ/sin θ = sin θ.

Example: If tan θ = 3/4 and θ is in Quadrant I, find sin θ and cos θ.
Draw a 3-4-5 triangle: opposite = 3, adjacent = 4, hypotenuse = 5.
sin θ = 3/5, cos θ = 4/5.

Law of Sines and Law of Cosines

Example 1 (Law of Cosines): A triangle has sides a = 8, b = 6, and angle C = 60°. Find side c.
c² = 64 + 36 − 2(8)(6)cos 60° = 100 − 96(0.5) = 100 − 48 = 52
c = √52 = 2√13 ≈ 7.21

Example 2 (Law of Sines): In triangle ABC, angle A = 40°, angle B = 75°, and side a = 10. Find side b.
Angle C = 180° − 40° − 75° = 65°. By Law of Sines: 10/sin 40° = b/sin 75°.
b = 10 × sin 75° / sin 40° = 10 × 0.966 / 0.643 ≈ 15.02.

Example 3 (Law of Cosines — finding an angle): A triangle has sides 5, 7, and 9. Find the largest angle.
The largest angle is opposite the longest side (9). 9² = 5² + 7² − 2(5)(7)cos C.
81 = 25 + 49 − 70 cos C → 81 = 74 − 70 cos C → cos C = −7/70 = −0.1.
C = cos⁻¹(−0.1) ≈ 95.7°.

Graphing Trig Functions

For y = A sin(Bx + C) + D: amplitude = |A|, period = 2π/|B|, phase shift = −C/B, vertical shift = D. The same applies to cosine. Tangent has period π/|B| and no amplitude.

Example: For y = −2cos(πx/3) + 5, find the amplitude, period, and range.
Amplitude = |−2| = 2. Period = 2π/(π/3) = 6. Vertical shift = 5.
Range: [5 − 2, 5 + 2] = [3, 7]. The negative sign flips the graph but does not change amplitude or range.

Inverse Trig Functions

sin⁻¹(x) (arcsin) returns values in [−π/2, π/2]. cos⁻¹(x) (arccos) returns values in [0, π]. tan⁻¹(x) (arctan) returns values in (−π/2, π/2). These "undo" the trig function to find the angle.

Example: Find sin⁻¹(√3/2).
Since sin(60°) = √3/2 and 60° is in [−90°, 90°], sin⁻¹(√3/2) = 60° = π/3.

Example: Find cos⁻¹(−1/2).
cos(120°) = −1/2 and 120° is in [0°, 180°], so cos⁻¹(−1/2) = 120° = 2π/3.

Common Mistakes

Quick Reference — Formula Sheet

ACT Strategies for Trigonometry

• Memorize the ratios: 30-60-90 and 45-45-90 side ratios appear constantly on the ACT.
• Sketch a triangle: If you forget a trig value, draw a right triangle with the known sides and read off the ratio.
• Non-right triangle? Think Law of Sines or Law of Cosines immediately.
• ASTC rule: Quickly determine sign (+/−) of any trig value by identifying the quadrant.
• Graphing shortcut: On graphing questions, identify amplitude and period first — these alone eliminate 2-3 answers.
• Convert freely: If a problem gives radians but you think better in degrees, convert right away.
• Identity substitution: If a problem looks like it needs simplification, try sin²θ + cos²θ = 1 first.
• Draw the reference angle: For any angle, find the reference angle first, compute the trig value, then apply the correct sign for the quadrant.
• Angle of elevation/depression: Always draw the horizontal line and the line of sight — the angle is between them.

Practice Walkthrough

ACT-Style Problem: In a right triangle, one leg has length 8 and the hypotenuse has length 17. What is the value of tan θ, where θ is the angle opposite the leg of length 8?

A. 8/17    B. 8/15    C. 15/17    D. 15/8

Step 1 — Find the missing leg: a² + 8² = 17² → a² = 289 − 64 = 225 → a = 15. (Recognize the 8-15-17 Pythagorean triple!)
Step 2 — Apply TOA: tan θ = Opposite/Adjacent = 8/15.
Elimination: Choice A (8/17) is sin θ (opposite/hypotenuse). Choice C (15/17) is cos θ (adjacent/hypotenuse). Choice D (15/8) is the reciprocal of the correct answer (cot θ or tan of the OTHER angle). Only B (8/15) is correct.