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

Use of \(x\) and \(y\) coordinates of points on the unit circle to give cosine and sine respectively.

The sine and cosine rules, including the ambiguous case of the sine rule.

The area of a triangle in the form \(\frac{1}{2}ab\sin{C}\).

Work with radian measure, including use for arc length and area of sector.

Use of the formulae \(s = rθ\) and \(A = \frac{1}{2}r^2θ\) for arc lengths and areas of sectors of a circle.

Trigonometric ratios Sine and cosine rules, area of a triangle Radians Arc length, area of a sector

Understand and use the standard small angle approximations of sine, cosine and tangent

\(\sin{θ}≈θ\), \(\cos{θ}≈1-\dfrac{θ^2}{2}\), \(\tan{θ}≈θ\)

Where θ is in radians.

Students should be able to approximate,

e.g. \(\dfrac{\cos{3x}-1}{x\sin{4x}}\) when \(x\) is small, to \(-\dfrac{9}{8}\).

Small angle approximations

Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity.

Knowledge of graphs of curves with equations such as \(y = \sin{x}\), \(y =\cos{(x + 30°)}\), \(y =\tan{2x}\) is expected.

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

Trigonometric graphs, exact values

Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains.

Angles measured in both degrees and radians.

Reciprocal trigonometric functions Inverse trigonometric functions

Understand and use

\(\tan{θ} = \dfrac{\sin{θ}}{\cos{θ}}\)

\(\sin^2{θ} + \cos^2{θ} = 1\)

\(\sec^2{θ} = 1 + \tan^2{θ}\) and

\(\cosec^2{θ} = 1 + \cot^2{θ}\)

These identities may be used to solve trigonometric equations and angles may be in degrees or radians. They may also be used to prove further identities.

Trigonometric identities

Understand and use double angle formulae; use of formulae for \(\sin{(A ± B)}\), \(\cos{(A ± B)}\), and \(\tan{(A ± B)}\), understand geometrical proofs of these formulae.

To include application to half angles.
Knowledge of the \(\tan{(\frac{1}{2}θ)}\) formulae will not be required.

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

Students should be able to solve equations such as \(a\cos{θ} + b\sin{θ} = c\) in a given interval.

Addition and double angle formulae Half angle and power reduction formulae a cosθ + b sinθ

Solve simple trigonometric equations in a given interval, including quadratic equations in \(\sin\), \(\cos\) and \(\tan\) and equations involving multiples of the unknown angle.

Students should be able to solve equations such as

\(\sin{(x + 70°)} = 0.5\) for \(0 < x < 360°\),

\(3 + 5\cos{2x} = 1\) for \(-180° < x < 180°\), and

\(6\cos^2{x} + \sin{x} − 5 = 0\) for \(0 ≤ x < 360°\)

These may be in degrees or radians and this will be specified in the question.

Solve trigonometric equations

Construct proofs involving trigonometric functions and identities.

Students need to prove identities such as
\(\cos{x}\cos{2x} + \sin{x}\sin{2x} ≡ \cos{x}\).

Proofs involving trigonometry

Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.

Problems could involve (for example) wave motion, the height of a point on a vertical circular wheel, or the hours of sunlight throughout the year. Angles may be measured in degrees or in radians.

Solve problems using trigonometry