Circular (trigonometric) functions — A/L Combined Mathematics

What must I remember about Circular (trigonometric) functions?

• Radian measure: Angle subtended at circle center by arc of length equal to radius: π radians = 180°; arc length s = rθ and sector area A = (1/2)r² θ (θ in radians).
• Trigonometric functions on unit circle: For angle θ, cos θ = x-coordinate, sin θ = y-coordinate; tan θ = sin θ / cos θ; sec θ = 1 / cos θ; cosec θ = 1 / sin θ; cot θ = 1 / tan θ.
• Fundamental Pythagorean identities: sin² θ + cos² θ = 1; 1 + tan² θ = sec² θ; 1 + cot² θ = cosec² θ.
• Compound Angle formulas: 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).
• Double and Triple Angle formulas: sin 2A = 2 sin A cos A; cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A; tan 2A = (2 tan A)/(1 - tan² A); sin 3A = 3 sin A - 4 sin³ A; cos 3A = 4 cos³ A - 3 cos A.

Test yourself (1 minute)

1. Product-to-Sum and Sum-to-Product conversions

   2 sin A cos B = sin(A + B) + sin(A - B); 2 cos A cos B = cos(A + B) + cos(A - B); 2 sin A sin B = cos(A - B) - cos(A + B); sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2); cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2).

2. t-substitution (Half-angle tangent substitution)

   Let t = tan(θ/2); then sin θ = 2t/(1 + t²), cos θ = (1 - t²)/(1 + t²), and tan θ = 2t/(1 - t²); converts trigonometric equations into algebra.

3. General solutions of trigonometric equations

   sin θ = sin α => θ = nπ + (-1)ⁿ α; cos θ = cos α => θ = 2nπ ± α; tan θ = tan α => θ = nπ + α (where n ∈ ℤ).

Practise Circular (trigonometric) functions questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 9

What comes next?

Next in the syllabus

Where does Circular (trigonometric) functions fit in the syllabus?

Year 1 · Term 2