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LearnAQA7357First teaching 2017Paper 1 · PureTrigonometry

Paper 1 · Module

Trigonometry

12 subtopics≈ 500 min in total

Start the module: Reciprocal functions and their identities Open graphing calculator

Reciprocal functions and their identities
Three new functions that are just $1$ over the old ones: their values, their graphs with asymptotes, the two Pythagorean identities they satisfy, and how to use them to solve equations.
≈ 40 min
Radians
What a radian is, why mathematicians prefer it to the degree, and how to convert between the two, with exact values in terms of $\pi$ .
≈ 30 min
Exact values and the unit circle
How sine, cosine and tangent are defined for any angle using the unit circle, the CAST rule for signs, and the exact values at $30^\circ$ , $45^\circ$ and $60^\circ$ that you need without a calculator.
≈ 40 min
Compound-angle formulae
$\sin(A+B)$ is not $\sin A+\sin B$ . The correct formulae for $\sin(A\pm B)$ , $\cos(A\pm B)$ and $\tan(A\pm B)$ , how to use them for exact values, and how they turn awkward equations into $\tan\theta=k$ .
≈ 45 min
Arc length and sector area
Why the formulae $s=r\theta$ and $A=\tfrac12r^2\theta$ work, how to use them for perimeters, and how to work backwards from a length or an area to the radius or angle.
≈ 35 min
Graphs and transformations
The shapes of $y=\sin x$ , $y=\cos x$ and $y=\tan x$ , their symmetries, and how $a$ , $b$ , $c$ and $d$ in $y=a\sin(b(x-c))+d$ stretch and shift them. Then use the graphs to count solutions and model things that repeat.
≈ 45 min
Segments, chords and composite shapes
Cutting a sector with a chord: the segment formula, chord lengths, and the exam-style shaded regions built from sectors, triangles and tangents.
≈ 45 min
Inverse trigonometric functions
Why $\sin x=\tfrac12$ has infinitely many solutions but $\sin^{-1}\tfrac12$ has only one, the restricted domains that make sine, cosine and tangent one-to-one, and the principal values your calculator gives.
≈ 35 min
Double-angle formulae
Setting $B=A$ in the compound-angle formulae gives formulae for $\sin2A$ , $\cos2A$ and $\tan2A$ . Three versions of $\cos2A$ let you match the function in an equation, and they are essential for integration later.
≈ 45 min
Trigonometric identities
The two identities at the heart of AS trigonometry, where they come from, and how to use them to simplify expressions, find exact values and write clear proofs.
≈ 40 min
The R-form (harmonic form)
Any combination $a\sin\theta+b\cos\theta$ is a single sine wave in disguise: $R\sin(\theta+\alpha)$ . How to find $R$ and $\alpha$ , and how this makes equations and maximum and minimum questions easy.
≈ 45 min
Solving trigonometric equations
How to find every solution of a trig equation in an interval: principal value, symmetry, and period. Then equations with $2x$ or $x+30^\circ$ inside, quadratics in $\sin x$ or $\cos x$ , and equations that need an identity first.
≈ 55 min