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Math: Trigonometry for the ACT Exam

Trigonometry is a small but consistent part of ACT Math and tends to concentrate in the harder questions (numbers 45–60). Despite the small question count, trig is disproportionately missed because many students study it last and encounter it least in their high school curriculum before test day. The ACT trig questions are straightforward if you know the fundamentals: SOHCAHTOA, reciprocal trig functions (csc, sec, cot), the Pythagorean identity (sin²θ + cos²θ = 1), and basic properties of sine and cosine graphs.

ACT Inc. — ACT Test Specifications: Mathematics section content areas and question distribution.

Locale-specific study guides

Pass-rate data, regulatory context, and study tips for Math: Trigonometry all change by candidate locale. Pick your context:

Common failure modes

These are the patterns that cause most candidates to lose marks on this topic. Recognising them in advance is half the work.

  • !Confusing sin, cos, and tan relationships with the wrong sides — always redraw SOHCAHTOA before answering (Opposite, Adjacent, Hypotenuse relative to the angle)
  • !Not knowing the reciprocal identities: csc = 1/sin, sec = 1/cos, cot = 1/tan
  • !Confusing radians and degrees on unit circle questions — the ACT uses both and often mixes them in the same question
  • !Forgetting that the range of sine and cosine is [−1, 1] — answers outside this range are impossible and signal a calculation error

Study tips

  • 1Memorize SOHCAHTOA and the reciprocal identities cold: sin=O/H, cos=A/H, tan=O/A; csc=H/O, sec=H/A, cot=A/O.
  • 2Learn the special angle values at 0°, 30°, 45°, 60°, 90°: sin and cos at these five angles are tested directly. Build a table and memorize it.
  • 3Practice converting between degrees and radians: multiply by π/180 to convert degrees to radians; multiply by 180/π to convert radians to degrees.
  • 4For sine and cosine graph questions, know: period of y = sin(bx) is 2π/b; amplitude of y = a·sin(x) is |a|; vertical shift y = sin(x) + c shifts the graph up c units.

Sample ACT Math: Trigonometry questions

These sample items mirror the format and difficulty of real ACT questions. Practice with thousands more on the free Koydo question bank.

  1. 1

    In a right triangle, the hypotenuse is 10 and one leg is 6. What is the cosine of the angle opposite the leg of length 6?

    • A3/5
    • B4/5Correct
    • C3/4
    • D6/10
    Why this answer?

    First find the missing leg: 6² + b² = 10² → b² = 100 − 36 = 64 → b = 8. The angle opposite the leg of 6 has: opposite = 6, adjacent = 8, hypotenuse = 10. cos(θ) = adjacent/hypotenuse = 8/10 = 4/5. Note: option D (6/10) would be sin(θ), not cos(θ) — the most common error here is applying SOHCAHTOA to the wrong angle. (Illustrative.)

  2. 2

    Which of the following is equivalent to sin²θ + cos²θ?

    • A0
    • B2
    • C1Correct
    • Dtan²θ
    Why this answer?

    sin²θ + cos²θ = 1 is the fundamental Pythagorean identity. It holds for all values of θ. This identity is derived from the Pythagorean theorem applied to the unit circle: for any point (cosθ, sinθ) on the unit circle, x² + y² = r² = 1.

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