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Math · Trigonometry

Hard Trigonometry Drill

15 hard-difficulty ACT Math questions on Trigonometry. Each question is on its own page with a worked answer explanation, the underlying rule, and a quick study tip.

Hard Math Trigonometry

All 15 questions

  1. 1 Sine of a Special Angle What is sin(45°)? Hard
  2. 2 SOHCAHTOA — Sine in a Right Triangle In a right triangle, the side opposite angle θ has length 5 and the hypotenuse has length Hard
  3. 3 Pythagorean Identity What is the value of sin²(θ) + cos²(θ) for any angle θ? Hard
  4. 4 Tangent in a Right Triangle In a right triangle, the side opposite angle θ has length 3 and the side adjacent to θ has Hard
  5. 5 Sine of a Special Angle What is sin(60°)? Hard
  6. 6 SOHCAHTOA — Sine in a Right Triangle In a right triangle, the side opposite angle θ has length 3 and the hypotenuse has length Hard
  7. 7 Pythagorean Identity What is the value of sin²(θ) + cos²(θ) for any angle θ? Hard
  8. 8 Tangent in a Right Triangle In a right triangle, the side opposite angle θ has length 9 and the side adjacent to θ has Hard
  9. 9 Sine of a Special Angle What is sin(30°)? Hard
  10. 10 SOHCAHTOA — Sine in a Right Triangle In a right triangle, the side opposite angle θ has length 8 and the hypotenuse has length Hard
  11. 11 Pythagorean Identity What is the value of sin²(θ) + cos²(θ) for any angle θ? Hard
  12. 12 Tangent in a Right Triangle In a right triangle, the side opposite angle θ has length 9 and the side adjacent to θ has Hard
  13. 13 Sine of a Special Angle What is sin(45°)? Hard
  14. 14 SOHCAHTOA — Sine in a Right Triangle In a right triangle, the side opposite angle θ has length 3 and the hypotenuse has length Hard
  15. 15 Pythagorean Identity What is the value of sin²(θ) + cos²(θ) for any angle θ? Hard

How to use this drill

Open the questions one at a time. Read the prompt, decide on an answer before scrolling, then check the explanation. The point of a drill is not to "get the question right" — it's to internalize the rule so you can apply it cold under timed conditions on test day.

If you're missing more than a third of the questions in this set, drop down a difficulty tier and rebuild from there. If you're getting all of them quickly, jump up a tier and try a mixed drill across subtopics.

What "Hard" means here

Hard questions are designed to separate strong test-takers. They typically require multiple rules applied in sequence, or careful elimination among answers that all look plausible. Expect to see these appear later in real ACT sections; even strong test-takers miss a couple per test.

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