Trigonometry — Edexcel A-Level Mathematics

For small angles measured in radians, trigonometric functions can be replaced by simple polynomial approximations: sin θ ≈ θ, cos θ ≈ 1 − θ²/2 and tan θ ≈ θ. These arise from the behaviour of the functions near zero. For example, if θ = 0.1 rad, then sin 0.1 ≈ 0.0998, which is close to 0.1; cos 0.1 ≈ 0.9950, matching 1 − 0.1²/2 = 0.995; and tan 0.1 ≈ 0.1003, close to 0.1. The approximations are used to simplify expressions, to find limiting values as θ → 0, and to model physical situations. They are only valid when θ is in radians and sufficiently small; the error grows as θ increases. Students should be able to substitute, simplify and compare exact and approximate values, and recognise when the approximation is appropriate.

The sine and cosine functions are defined for all real angles, while tangent is undefined at π/2 + kπ for integer k. Their graphs show characteristic symmetries and periodicities. Sine and cosine have period 2π, while tangent has period π. Sine and tangent are odd functions, whereas cosine is even. Exact values at key angles allow precise work without a calculator: sin 0 = 0, sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2, sin(π/2) = 1, sin π = 0; cos 0 = 1, cos(π/6) = √3/2, cos(π/4) = √2/2, cos(π/3) = 1/2, cos(π/2) = 0, cos π = −1; tan 0 = 0, tan(π/6) = 1/√3, tan(π/4) = 1, tan(π/3) = √3, tan π = 0. Multiples follow from periodicity and symmetry, for example sin(5π/6) = 1/2, cos(4π/3) = −1/2, tan(7π/6) = 1/√3. Students should sketch graphs, identify symmetries, use periodicity, and solve equations using exact values.

Secant, cosecant and cotangent are reciprocal trigonometric functions: sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ = cos θ/sin θ. Their domains exclude values where the denominator is zero, so sec θ and tan θ are undefined where cos θ = 0, and cosec θ and cot θ are undefined where sin θ = 0. Their graphs show vertical asymptotes at these excluded values. The inverse trigonometric functions arcsin, arccos and arctan return an angle for a given ratio. Because sine and cosine are periodic, their inverses are restricted to principal ranges: arcsin x has domain [−1, 1] and range [−π/2, π/2]; arccos x has domain [−1, 1] and range [0, π]; arctan x has domain ℝ and range (−π/2, π/2). Students should sketch these graphs, state domains and ranges, and use relationships such as sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ.

The identity tan θ = sin θ / cos θ defines tangent in terms of sine and cosine and is valid wherever cos θ ≠ 0. The Pythagorean identity sin²θ + cos²θ = 1 holds for all θ and is the foundation for the other two identities. Dividing sin²θ + cos²θ = 1 by cos²θ gives tan²θ + 1 = sec²θ, usually written sec²θ = 1 + tan²θ, valid where cos θ ≠ 0. Dividing by sin²θ gives 1 + cot²θ = cosec²θ, usually written cosec²θ = 1 + cot²θ, valid where sin θ ≠ 0. These identities are used to simplify expressions, prove other identities, and solve trigonometric equations. For example, to solve 2sin²θ + 3cos²θ = 2, replace sin²θ with 1 − cos²θ to obtain 2 + cos²θ = 2, so cos²θ = 0 and θ = π/2 + kπ. Students should manipulate expressions fluently and state any restrictions on θ.

Addition formulae are sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B, and tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B). Setting B = A gives double angle results like sin 2A = 2 sin A cos A. Geometrical proofs of addition formulae use stacked right-angled triangles: a triangle with angle B sits on the hypotenuse of one with angle A. Resolving sides derives sin(A + B) and cos(A + B). The harmonic form a cos θ + b sin θ (for a, b > 0) can be expressed as R cos(θ − α) where R = √(a² + b²) and tan α = b/a, or R sin(θ + α) where tan α = a/b. For example, 3 cos θ + 4 sin θ = 5 cos(θ − 53.1°) since R = √(3²+4²) = 5 and tan α = 4/3. Other sign combinations map to R cos(θ + α) or R sin(θ − α).

Solving trigonometric equations requires finding all angles in a specified interval that satisfy an equation. For simple equations like sin θ = k, use the inverse function to find the principal value, then use symmetry and periodicity of the trigonometric functions to find all solutions in the interval. For quadratic equations in sin θ, cos θ or tan θ, rearrange to standard form, factorise or use the quadratic formula, then solve each resulting simple equation. Equations involving multiples of the unknown angle, such as sin 2θ = k or cos(3θ + 30°) = k, are solved by first finding the general solution for the multiple angle, then dividing or adjusting to find θ in the required interval. Always check that solutions lie within the given interval and consider the periodic nature of the functions.

Constructing proofs involving trigonometric functions and identities requires manipulating expressions using known identities to show that one side of an equation is equivalent to the other. Start with the more complicated side and simplify using fundamental identities such as sin² θ + cos² θ = 1, tan θ = sin θ / cos θ, and the addition, double angle and harmonic form formulae. Techniques include factorising, combining fractions, multiplying numerator and denominator by a conjugate, and substituting equivalent expressions. For example, to prove that (1 − cos 2θ) / sin 2θ = tan θ, use cos 2θ = 1 − 2 sin² θ and sin 2θ = 2 sin θ cos θ, then simplify. Proofs must be logically sequenced, with each step justified by a named identity or algebraic operation, and must not assume the result being proved.

Trigonometric functions model real-world problems involving vectors, kinematics and forces. In vector problems, resolve vectors into components using sine and cosine, or find angles using the cosine rule on a vector triangle. In kinematics, trigonometric equations describe projectile motion where horizontal and vertical components are analysed separately, or particles on inclined planes. For forces, resolve into perpendicular components using sine and cosine, and find resultants using vector addition or the cosine rule. For example, a force of 10 N at 30° to the horizontal has a horizontal component of 10 cos 30° = 5√3 ≈ 8.66 N and a vertical component of 10 sin 30° = 5 N. Always draw a clear diagram, define the positive direction, and use exact values where possible.