Example 1: Solving a trigonometric equation on 0 to 2π
Solve sin 2x = cos x for 0 ≤ x ≤ 2π.
2 sin x cos x − cos x = 0cos x (2 sin x − 1) = 0cos x = 0givesx = π/2, 3π/2sin x = ½givesx = π/6, 5π/6
Answer: x = π/6, π/2, 5π/6, 3π/2
Trigonometry in DSE M2: Compound Angle, Double Angle and Sum-to-Product Formulas
M2 adds four things to Core trigonometry: radian measure, the functions sec, cosec and cot, the compound and double angle formulas such as sin(A + B) = sin A cos B + cos A sin B, and the product-to-sum and sum-to-product formulas. They exist mainly to make differentiation and integration of trigonometric functions possible.
Trigonometric Formulas (DSE M2)
Radians
π rad = 180°Identities
1 + tan² θ = sec² θ, 1 + cot² θ = cosec² θCompound angle (sine)
sin(A ± B) = sin A cos B ± cos A sin BCompound angle (cosine)
cos(A ± B) = cos A cos B ∓ sin A sin BCompound angle (tangent)
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)Double angle
sin 2A = 2 sin A cos A, cos 2A = cos² A − sin² A = 1 − 2 sin² A = 2 cos² A − 1Power reduction
sin² A = ½(1 − cos 2A), cos² A = ½(1 + cos 2A)Product to sum
2 sin A cos B = sin(A + B) + sin(A − B), 2 cos A cos B = cos(A + B) + cos(A − B)Sum to product
sin A + sin B = 2 sin((A + B)/2) cos((A − B)/2), cos A + cos B = 2 cos((A + B)/2) cos((A − B)/2)Worked Examples — More about Trigonometric Functions
Example 2: Proving an identity with sum-to-product
Prove that (sin 3x + sin x) / (cos 3x + cos x) = tan 2x.
sin 3x + sin x = 2 sin 2x cos xcos 3x + cos x = 2 cos 2x cos x- L.H.S.
= (2 sin 2x cos x) / (2 cos 2x cos x) = sin 2x / cos 2x = tan 2x= R.H.S.
Answer: Proved.
💡 Exam Traps & Secrets
⚠️ Do not divide by cos xIn Example 1, dividing both sides by
cos x throws away the solutions π/2 and 3π/2. Move everything to one side and factorise instead.⚠️ Radians in calculusAll M2 calculus with trigonometric functions assumes radians. Give solutions as multiples of π unless degrees are asked for, and set your calculator to radian mode.
⚠️ Subsidiary angle form is not requiredWriting
a sin x + b cos x as R sin(x + α) is not in the M2 syllabus, and neither are general solutions outside 0 to 2π.✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Compound angle)Without a calculator, find the exact value of sin 75°.
(Double angle)Prove that (1 − cos 2x)/sin 2x = tan x.
(Equation with double angle)Solve cos 2x + 3 sin x = 2 for 0 ≤ x ≤ 2π.
Frequently Asked Questions
sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B; tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B). Putting B = A gives the double angle formulas.
Mainly for integration. A product such as sin 3x cos x cannot be integrated directly, but written as ½(sin 4x + sin 2x) it can. They are also used to prove identities and solve equations.
No. The curriculum notes state that the subsidiary angle form is not required.