Practice Maths

Trigonometry and Functions — Topic Review

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This review covers all four lessons in the Trigonometry and Functions topic: Pythagorean and Reciprocal Identities, Compound and Double Angle Formulas, Trigonometric Equations with Identities, and Inverse Trigonometric Functions. Questions are exam-style and increase in difficulty.

Mixed Review Questions

  1. Fluency

    Q1 — Pythagorean Identity

    Simplify each expression using the Pythagorean identity sin²θ + cos²θ = 1:

    (a) 1 − sin²θ    (b) sin²θ − 1    (c) 1 − 2cos²θ + cos4θ

  2. Fluency

    Q2 — Reciprocal Identities

    If tanθ = 3/4 and θ is in the first quadrant, find the exact values of: (a) secθ    (b) cosecθ    (c) cotθ

  3. Fluency

    Q3 — Exact Value Using Compound Angle Formula

    Find the exact value of sin 75°, using the compound angle formula for sin(A + B).

  4. Fluency

    Q4 — Exact Value Using Compound Angle Formula

    Find the exact value of cos 15°.

  5. Fluency

    Q5 — Double Angle Formula

    Given sinθ = 5/13 and θ is in the first quadrant, find the exact values of: (a) sin(2θ)    (b) cos(2θ)

  6. Understanding

    Q6 — Proving a Trig Identity

    Prove that (sinθ + cosθ)² = 1 + sin(2θ).

  7. Understanding

    Q7 — Solving a Trig Equation with Identities

    Solve 2sin²θ − sinθ − 1 = 0 for θ ∈ [0°, 360°].

  8. Understanding

    Q8 — Using a Double Angle Identity to Solve an Equation

    Solve sin(2θ) = sinθ for θ ∈ [0, 2π].

  9. Understanding

    Q9 — Inverse Trig Values

    Find the exact value of: (a) arcsin(1/2)    (b) arctan(√3)    (c) arccos(−1/2)    (d) arctan(−1)

  10. Understanding

    Q10 — Solving an Inverse Trig Equation

    Solve: (a) arccos(x) = π/3    (b) arctan(x) = π/4    (c) arcsin(2x − 1) = π/6

  11. Understanding

    Q11 — Proving with Reciprocal Identities

    Prove that tan²θ + 1 = sec²θ.

  12. Problem Solving

    Q12 — Complex Equation Using Multiple Identities

    Solve cos(2θ) + cosθ = 0 for θ ∈ [0, 2π].

  13. Problem Solving

    Q13 — Reciprocal Identity Proof

    Prove that (cosecθ − cotθ)(cosecθ + cotθ) = 1.

  14. Problem Solving

    Q14 — Compound Angle — Finding sin(A+B) Given Quadrant Conditions

    Given sin A = 3/5 with A in Q1, and cos B = −5/13 with B in Q2, find the exact value of sin(A + B).

  15. Problem Solving

    Q15 — Domain and Range of Inverse Trig

    For f(x) = arcsin(x): (a) State the domain and range. (b) Sketch the graph. (c) Solve arcsin(x) = arccos(x) and explain geometrically.

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