Understand and be able to use the definitions of sine, cosine and tangent for all arguments.

Trigonometric ratios

Understand and be able to use the sine and cosine rules.

Questions may include the use of bearings and require the use of the ambiguous case of the sine rule.

Sine and cosine rules, area of a triangle

Understand and be able to use the area of a triangle in the form \(\frac{1}{2}ab\sin{C}\).

Sine and cosine rules, area of a triangle

Be able to work with radian measure, including use for arc length and area of sector.

Learners should know the formulae \(s = r\theta\) and \(A=\frac{1}{2}r^2\theta\).

Learners should be able to use the relationship between degrees and radians.

Radians Arc length, area of a sector

Understand and be able to use the standard small angle approximations of sine, cosine and tangent:

1. \(\sin{\theta} ≈ \theta\),


2. \(\cos{\theta} ≈ 1 - \dfrac{\theta^2}{2}\),


3. \(\tan{\theta} ≈ \theta\),

Small angle approximations

Understand and be able to use the sine, cosine and tangent functions, their graphs, symmetries and periodicities.

Includes knowing and being able to use exact values of \(\sin{\theta}\) and \(\cos{\theta}\) for \(\theta = 0°, 30°, 45°, 60°, 90°, 180°\) and multiples thereof and exact values of \(\tan{\theta}\) for \(\theta = 0°, 30°, 45°, 60°, 180°\) and multiples thereof.

Trigonometric graphs, exact values

Know and be able to use exact values of \(\sin{\theta}\) and \(\cos{\theta}\) for \(\theta = 0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \dfrac{\pi}{2}, \pi\) and multiples thereof, and exact values of \(\tan{\theta}\) for \(\theta = 0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \pi\) and multiples thereof.

Trigonometric graphs, exact values

Understand and be able to use the definitions of secant (\(\sec{\theta}\)), cosecant (\(\cosec{\theta}\)) and cotangent (\(\cot{\theta}\)) and of \(\arcsin{\theta}\), \(\arccos{\theta}\) and \(\arctan{\theta}\) and their relationships to \(\sin{\theta}\), \(\cos{\theta}\) and \(\tan{\theta}\) respectively.

Reciprocal trigonometric functions Inverse trigonometric functions

Understand the graphs of the functions given in 1.05h, their ranges and domains.

In particular, learners should know that the principal values of the inverse trigonometric relations may be denoted by \(\arcsin{\theta}\) or \(\sin^{-1}{\theta}\), \(\arccos{\theta}\) or \(\cos^{-1}{\theta}\), \(\arctan{\theta}\) or \(\tan^{-1}{\theta}\) and relate their graphs (for the appropriate domain) to the graphs of \(\sin{\theta}\), \(\cos{\theta}\) and \(\tan{\theta}\).

Reciprocal trigonometric functions Inverse trigonometric functions

Understand and be able to use \(\tan{θ} ≡ \dfrac{\sin{θ}}{\cos{θ}}\) and \(\sin^2{θ} + \cos^2{θ} ≡ 1\).

In particular, these identities may be used in solving trigonometric equations and simple trigonometric proofs.

Trigonometric identities

Understand and be able to use \(\sec^2{θ} ≡ 1 + \tan^2{θ}\) and \(\cosec^2{θ} ≡ 1 + \cot^2{θ}\).

In particular, the identities in 1.05j and 1.05k may be used in solving trigonometric equations, proving trigonometric identities or in evaluating integrals.

Trigonometric identities

Understand and be able to use double angle formulae and the formulae for \(\sin{(A ± B)}\), \(\cos{(A ± B)}\), and \(\tan{(A ± B)}\).

Learners may be required to use the formulae to prove trigonometric identities, simplify expressions, evaluate expressions exactly, solve trigonometric equations or find derivatives and integrals.

Addition and double angle formulae

Understand the geometrical proofs of these formulae.

Addition and double angle formulae

Understand and be able to use expressions for \(a\cos{θ} + b\sin{θ}\) in the equivalent forms of \(R\cos{(θ±α)}\) or \(R\sin{(θ±α)}\).

In particular, learners should be able to:
1. sketch graphs of \(a\cos{θ} + b\sin{θ}\),
2. determine features of the graphs including minimum or maximum points and
3. solve equations of the form \(a\cos{θ} + b\sin{θ} = c\).

a cosθ + b sinθ

Be able to solve simple trigonometric equations in a given interval, including quadratic equations in \(\sin{\theta}\), \(\cos{\theta}\) and \(\tan{\theta}\) and equations involving multiples of the unknown angle.

e.g.
\(\sin{θ} = 0.5\) for \(0 ≤ θ < 360°\)
\(6\sin^2{θ} + \cos{θ} - 4 = 0\) for \(0 ≤ θ < 360°\)
\(\tan{3θ} = -1\) for \(-180° < θ < 180°\)

Extend their knowledge of trigonometric equations to include radians and the trigonometric identities in Stage 2.

Solve trigonometric equations

Be able to construct proofs involving trigonometric functions and identities.

e.g. Prove that \(\cos^2{(θ+45°)} - \frac{1}{2}(\cos{2θ}-\sin{2θ}) = \sin^2{θ}\).

Includes constructing a mathematical argument as described in Section 1.01.

Proofs involving trigonometry

Be able to use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

Problems may include realistic contexts, e.g. movement of tides, sound waves, etc. as well as problems in vector form which involve resolving directions and quantities in mechanics.

Solve problems using trigonometry