A-Level Maths OCR A H240

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

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.

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

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.*

Understand and be able to use the standard small angle approximations of sine, cosine and tangent: olli$\sin{\theta} ≈ \theta$,/lili$\cos{\theta} ≈ 1 - \dfrac{\theta^2}{2}$,/lili$\tan{\theta} ≈ \theta$,[/li]/ol where $\theta$ is in radians.

e.g. Find an approximate expression for $\dfrac{\sin{3\theta}}{1+\cos{\theta}}$ if $\theta$ is small enough to neglect terms in $\theta^3$ or above.

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.

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.

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.

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 $\tan{\theta}$.

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.

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.

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.

Understand the geometrical proofs of these 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 $a\cos{θ} + b\sin{θ} = c$.*

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.*

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.*

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.