Pure Mathematics 3A Level

3.3 Trigonometry

2 of 2 objectives have lessons · 0 studied · 0 secure

Lessons

Sec, cosec and cot: the reciprocal threeThree new names, all built from the functions you already have — and each graph is just the inverse of one you know. 8 min, not started
The Pythagorean identities in disguiseDivide sin²θ + cos²θ ≡ 1 by cos²θ or sin²θ and the two new identities appear — and they turn mixed sec/tan equations into quadratics. 8 min, not started
sin(A ± B) and friends: exact values without a calculatorThe addition formulae let you compute trig functions of awkward angles from nice ones — and they are the engine behind the double-angle and R-form results. 8 min, not started
Double angles: sin 2A, cos 2A and tan 2APut B = A in the addition formulae and you get three new identities — cos 2A in particular has three interchangeable versions, and picking the right one is the whole skill. 8 min, not started
The R-form: a sin θ + b cos θ in one functionAny mixture a sin θ + b cos θ can be written as R sin(θ + α). That one move gives the maximum and minimum instantly and turns hard equations into easy ones. 9 min, not started

Syllabus objectives

2 objectives

What Cambridge says you should be able to do, in our words. Codes are the topic number plus the objective’s position.

3.3.1

Understand how the secant, cosecant and cotangent functions relate to cosine, sine and tangent, and use the properties and graphs of all six trigonometric functions for angles of any magnitude.

Taught in: Sec, cosec and cot: the reciprocal three · 5 questions

3.3.2

Use trigonometrical identities to simplify and exactly evaluate expressions and in the course of solving equations, selecting an identity or identities appropriate to the context; in particular use sec²θ ≡ 1 + tan²θ and cosec²θ ≡ 1 + cot²θ, the expansions of sin(A ± B), cos(A ± B) and tan(A ± B), the formulae for sin 2A, cos 2A and tan 2A, and the expression of a sin θ + b cos θ in the forms R sin(θ ± α) and R cos(θ ± α).

Examples: simplifying cos(x − 30°) − 3 sin(x − 60°); solving tan θ + cot θ = 4, 2 sec²θ − tan θ = 5 and 3 cos θ + 2 sin θ = 1.

Taught in: sin(A ± B) and friends: exact values without a calculator, Double angles: sin 2A, cos 2A and tan 2A, The Pythagorean identities in disguise, The R-form: a sin θ + b cos θ in one function · 22 questions