Trigonometry - Functions & Identities

Formulae Families & Memory Work

Your booklet provides 24 identities, but you must know how to navigate them: 1 (Pythagorean), 2-3 (Triangle rules), 4, 5, 7, 8, 9, 10 (Compound angles), 6 and 11-14 (Double angles), 15-16 (Squared forms/power reduction), and 17-24 (Product-to-sum/sum-to-product). You do not need to memorise the formulas themselves, but you must know their structures by heart to find them quickly. Crucially, you must memorise: the shapes and periods of sine, cosine, and tangent graphs; the CAST rule; exact surd values for 0°, 30°, 45°, 60°, 90°; and the logical steps to derive formulae 1, 2, 3, 4, 5, 6, 7, and 9.

Mandatory Formula Derivations

Derivations must follow the order in the booklet. For cos(A-B) (Formula 4): Use the unit circle, centre O, radius 1. Let P=(cos A, sin A) and Q=(cos B, sin B). |PQ|² = (cos A - cos B)² + (sin A - sin B)² = 2 - 2(cos A cos B + sin A sin B). By the cosine rule, |PQ|² = 1² + 1² - 2(1)(1)cos(A-B) = 2 - 2cos(A-B). Equating them gives cos(A-B) = cos A cos B + sin A sin B. For tan(A+B) (Formula 9): tan(A+B) = sin(A+B)/cos(A+B) = (sin A cos B + cos A sin B) / (cos A cos B - sin A sin B). Divide every term by cos A cos B to get (tan A + tan B) / (1 - tan A tan B). For tan(A-B), replace B with -B.

Solving Trigonometric Equations

Follow this recipe: 1. Isolate the trigonometric term. 2. Use identities (like Formula 1 or 11) to get the same ratio and same multiple of θ. 3. Find the reference angle α = inverse of the positive value. 4. Use CAST to pick the quadrants. 5. Write down all values for the multiple angle (e.g., 2θ), adding 360°n/2nπ to capture all solutions. 6. Divide by the coefficient. Rules for General Solutions (n ∈ ℤ): sin θ = sin α ⇒ θ = 360°n + α or θ = 360°n + (180° - α); cos θ = cos α ⇒ θ = 360°n ± α; tan θ = tan α ⇒ θ = 180°n + α (tan repeats every 180°).

Modelling Periodic Phenomena

For f(t) = a + b cos(ct) or a + b sin(ct): The central axis is a = (max + min)/2; the amplitude is |b| = (max - min)/2; the period is T = 2π/|c|, so c = 2π/T. If the model starts at a maximum at t = 0, use cosine. If it starts on the central axis, use sine. Always use RAD when differentiation or integration is involved or when the question mentions π.

The Tangent Graph & Identities

The graph of y = tan θ has vertical asymptotes at θ = π/2 + nπ and period π. It passes through (0,0), (π/4, 1), and (-π/4, -1). For 'Show that' identities, start with the more complex side. If you see sums of sines/cosines, use formulae 21-24: (cos 7A + cos A)/(sin 7A - sin A) becomes (2cos 4A cos 3A)/(2cos 4A sin 3A) = cot 3A.

Key terms

Amplitude

The distance from the central axis to the maximum or minimum of a wave, equal to |b|.

Period

The horizontal distance for one full wave cycle; 2π/|c| for sin/cos, π/|c| for tan.

Reference Angle

The acute angle formed with the x-axis, used to find values in other quadrants via CAST.

General Solution

A formula using an integer n to represent all possible angle solutions satisfying an equation.

Radian

An angle measure where π radians = 180°; essential for calculus applications.