Trigonometric equations — HL identities in a finite interval (IB Math AA HL)

Key ideas

• Do not divide by cos x or sin x: you lose the solutions where it is 0. Move everything to one side and factorise.
• Double angle: sin 2x = 2 sin x cos x, and cos 2x = 2cos² x − 1 = 1 − 2sin² x = cos² x − sin² x. Choose the version that leaves one function.
• Reciprocal identities: 1 + tan² x = sec² x and 1 + cot² x = cosec² x turn an equation into a quadratic in one function.
• For cos 3x = k with 0 ≤ x ≤ π, solve for 3x over 0 ≤ 3x ≤ 3π, then divide every answer by 3.
• Check the number of solutions against a sketch of the graph on the interval.

Worked example

(a) Solve sin 2x = cos x for 0 ≤ x ≤ 2π. (b) Solve cos 3x = 1/2 for 0 ≤ x ≤ π.

1. (a) Use sin 2x = 2 sin x cos x: 2 sin x cos x − cos x = 0.
2. Factorise: cos x (2 sin x − 1) = 0.
3. cos x = 0 gives x = π/2, 3π/2. sin x = 1/2 gives x = π/6, 5π/6.
4. (b) 0 ≤ x ≤ π means 0 ≤ 3x ≤ 3π. cos 3x = 1/2 gives 3x = π/3, 5π/3, 7π/3 (the next one, 11π/3, is too big).
5. Divide by 3: x = π/9, 5π/9, 7π/9.

Answer: (a) x = π/6, π/2, 5π/6, 3π/2. (b) x = π/9, 5π/9, 7π/9.

Exam-style HL AA questions on this topic

• Reciprocal and Inverse Trig Functions questions — 58 questions with mark schemes, plus free worked examples
• All HL AA Geometry and Trigonometry questions by topic

Do it on your GDC

A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Solving equations on your calculator

See it on a graph

• Graph of y = sin x
• Graph of y = tan x
• Intersections and turning points
• Open the free Grapher

Maths IA ideas that use trigonometric equations

• The hidden angles of a hip roof: 3D trigonometry
• What is the best angle to shoot a free throw?
• Where is the best seat in the cinema?
• Why does a random game draw the Sierpiński triangle?
• All HL AA IA ideas

Related in Geometry and Trigonometry

• 3D vectors — dot product, cross product, and applications
• Lines and planes in 3D — intersections and distances
• Vector proofs and geometric arguments

Revise Trigonometric equations for the exam

• HL AA revision notes — every unit summarised, with the key results to learn
• HL AA past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AA formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AA formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original geometry problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S71 · Problem S78 · Problem S21 · Problem S77. All 22 →

Sibling topic in other IB Maths courses