Compound and double-angle formulas (IB Math AA HL)

Key ideas

• sin(A ± B) = sin A cos B ± cos A sin B.
• cos(A ± B) = cos A cos B ∓ sin A sin B: the sign in the middle is the opposite of the sign in the bracket.
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B).
• Put B = A for the double-angle formulas: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan 2A = 2 tan A / (1 − tan²A).
• Exact values: split the angle into two you know, such as 75° = 45° + 30° or 15° = 45° − 30°.
• In a proof, start from one side and work to the other, and pick the form of cos 2A that cancels what you have: 2cos²A − 1 clears a +1, 1 − 2sin²A clears a −1.

Worked example

Find the exact value of sin 75°, then show that sin 2θ / (1 + cos 2θ) = tan θ.

1. sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°.
2. = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
3. For the identity, use cos 2θ = 2cos²θ − 1, so the 1 cancels: 1 + cos 2θ = 2cos²θ.
4. sin 2θ / (1 + cos 2θ) = 2 sin θ cos θ / (2cos²θ) = sin θ / cos θ = tan θ.

Answer: sin 75° = (√6 + √2)/4, and the left side simplifies to tan θ (for cos θ ≠ 0).

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

See it on a graph

• Graph of y = sec x
• Graph of y = cot x
• Open the free Grapher

Maths IA ideas that use compound and double-angle formulas

• 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

• Trigonometric equations — HL identities in a finite interval
• 3D vectors — dot product, cross product, and applications
• Lines and planes in 3D — intersections and distances

Revise Compound and double-angle formulas 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 S77 · Problem S20 · Problem S76 · Problem S19. All 22 →

Formulas in this topic: Trig identities. All formulas A–Z