E: Trigonometry — AQA A-Level Mathematics

For small angles measured in radians, the trigonometric functions behave almost like simple polynomials: sin θ ≈ θ, tan θ ≈ θ and cos θ ≈ 1 − θ²/2. These match the leading terms of the Maclaurin series, and they are accurate only when θ is small and in radians — using degrees breaks them completely. A useful check is θ = 0.1: sin 0.1 ≈ 0.099833, close to 0.1; cos 0.1 ≈ 0.995004, close to 1 − 0.005 = 0.995. To use them, replace each function by its approximation, simplify algebraically, and keep the leading terms. They are especially powerful for limits such as (1 − cos θ)/θ² → 1/2 as θ → 0, and for estimating values without a calculator.

The sine, cosine and tangent functions are defined for angles in radians and have characteristic graphs. Sin and cos oscillate between −1 and 1 with period 2π and amplitude 1; tan has period π, no amplitude, and vertical asymptotes where cos θ = 0, for example at θ = π/2. Symmetries include sin(π − θ) = sin θ, cos(−θ) = cos θ and tan(π + θ) = tan θ. Exact values follow from the special triangles: sin π/6 = 1/2, sin π/4 = √2/2, sin π/3 = √3/2, with cos values reversed, and tan π/6 = 1/√3, tan π/4 = 1, tan π/3 = √3. Multiples are handled using periodicity and symmetry, for example sin 5π/6 = sin π/6 = 1/2 and cos 4π/3 = −cos π/3 = −1/2. Note that tan π/2 is undefined, so it is excluded from the exact values required for tan.

This topic extends trigonometry beyond sine, cosine and tangent. Define the reciprocals: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = cos θ/sin θ (equivalently 1/tan θ where tan θ is defined and non-zero). The inverse functions arcsin, arccos and arctan return angles for given ratios. Graphs, domains and ranges are essential. Sec and cosec have vertical asymptotes where cos or sin are zero, with ranges |y| ≥ 1; cot has asymptotes where sin is zero and range all real numbers. The inverse graphs are the reflections of the restricted sine, cosine and tangent graphs in y = x: arcsin has domain [-1, 1] and range [-π/2, π/2]; arccos has domain [-1, 1] and range [0, π]; arctan has domain all real numbers and range (-π/2, π/2). Use these relationships to solve equations and prove identities.

This topic covers three fundamental Pythagorean identities in trigonometry. The first, sin²θ + cos²θ ≡ 1, is derived from the unit circle and holds for all θ. Dividing this identity by cos²θ gives sec²θ ≡ 1 + tan²θ, valid where cos θ ≠ 0. Dividing by sin²θ gives cosec²θ ≡ 1 + cot²θ, valid where sin θ ≠ 0. You must understand how these identities are derived and use them to simplify expressions, prove other identities, and solve trigonometric equations. For example, you can replace 1 + tan²θ with sec²θ to simplify an expression, or use sin²θ = 1 - cos²θ to convert between functions. These identities are essential tools for manipulating trigonometric expressions and are frequently examined in proofs and equation solving.

This topic covers compound angle formulae, double angle formulae, their geometrical proofs, and the harmonic form R cos(θ ± α) or R sin(θ ± α). You must derive and apply sin(A±B), cos(A±B), tan(A±B), and the double angle results for sin 2θ, cos 2θ, tan 2θ. Geometrical proofs often use a unit circle or right-angled triangles to establish these identities. For a cos θ + b sin θ, you rewrite it as R cos(θ ± α) or R sin(θ ± α), where R = √(a² + b²) and α satisfies cos α = a/R, sin α = b/R (for R cos(θ - α)). This form helps solve equations and find maxima/minima. For example, 3 cos θ + 4 sin θ = 5 cos(θ - 53.13°). You must handle signs and choose the correct form.

This topic requires solving trigonometric equations within a specified interval, such as 0° ≤ θ < 360° or 0 ≤ x < 2π. You must handle simple equations like sin θ = k, quadratic equations in sin, cos or tan (e.g., 2sin²θ - sinθ - 1 = 0), and equations with multiples of the angle (e.g., sin 2θ = 0.5). The method involves finding the principal value, using symmetry of trigonometric graphs to find all solutions in the interval, and adjusting for multiples by dividing the interval accordingly. For quadratic equations, factorise or use the quadratic formula, then solve each factor. Always check solutions are within the given interval and consider degrees or radians as specified.

This topic requires you to build logical arguments that establish trigonometric identities or solve equations by proof. You must use fundamental identities such as sin²θ + cos²θ ≡ 1, tanθ ≡ sinθ/cosθ, and the addition formulae, along with algebraic manipulation. A typical proof might ask you to show that (1 - cos²θ)/sinθ ≡ sinθ for all θ where defined. You would start with the left-hand side, replace 1 - cos²θ with sin²θ, then simplify to sinθ. Proofs must be clear, with each step justified, and you must avoid assuming the result. You also need to handle domains and restrictions, such as cosθ ≠ 0 when using tanθ. The skill is assessed through written responses where logical flow and correct use of identities are key.

This topic involves applying trigonometric functions to model and solve real-world problems. You might resolve a force into components using sine and cosine, analyse projectile motion by separating horizontal and vertical components, or find the resultant of vectors given as magnitudes and directions. For example, a force of 10 N at 30° to the horizontal has a horizontal component 10cos30° and a vertical component 10sin30°. You need to interpret the context, set up equations, and solve using trigonometry. Problems may require the sine or cosine rule in triangles, or using trigonometric identities to simplify. Assessment focuses on correct modelling, accurate calculations, and clear communication of the solution in context.