#7.2
Be able to use the definitions of \(\sin{θ}\), \(\cos{θ}\) and \(\tan{θ}\) for any angle.
By reference to the unit circle, \(\sin{θ}=y\), \(\cos{θ}=x\), \(\tan{θ}=\dfrac{y}{x}\).
#7.3
Know and use the graphs of \(\sin{θ}\), \(\cos{θ}\) and \(\tan{θ}\) for all values of \(θ\), their symmetries and periodicities.
Stretches, translations and reflections of these graphs.
Combinations of these transformations.
Notation: Period
#7.4
Know and be able to use the exact values of \(\sin{θ}\) and \(\cos{θ}\) for \(θ = 0°, 30°, 45°, 60°, 90°\) and the exact values of \(\tan{θ}\) for \(θ = 0°, 30°, 45°, 60°\).
#7.5
Know and be able to use the fact that the area of a triangle is given by \(\dfrac{1}{2}ab\sin{C}\).
#7.6
Know and be able to use the sine and cosine rules.
Use of bearings may be required.
#7.7
Understand and be able to use \(\tan{θ} = \dfrac{\sin{θ}}{\cos{θ}}\)
e.g. solve \(\sin{θ} = 3\cos{θ}\) for \(0° \le θ \le 360°\).
#7.8
Understand and be able to use the identity \(\sin^2{θ} + \cos^2{θ} = 1\).
e.g. solve \(\sin^2{θ} = \cos{θ}\) for \(0° \le θ \le 360°\).
#7.9
Be able to solve simple trigonometric equations in given intervals and know the principal values from the inverse trigonometric functions.
e.g. \(\sin{θ} = 0.5\), in \([0°, 360°] \iff θ = 30° ,150°\)
Includes equations involving multiples of the unknown angle e.g. \(\sin{2θ} = 3\cos{2θ}\).
Includes quadratic equations.
Notation:
\(\arcsin{x} = \sin^{-1}{x}\)
\(\arccos{x} = \cos^{-1}{x}\)
\(\arctan{x} = \tan^{-1}{x}\)
[Excludes: General solutions.]
#7.10
Know and be able to use exact values of \(\sin{θ}\), \(\cos{θ}\) and \(\tan{θ}\) for \(θ = 0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}\), \(\pi\) and multiples thereof and \(\sin{θ}\), \(\cos{θ}\) for \(θ = \dfrac{\pi}{2}\) and multiples thereof.
#7.11
Understand and use the definitions of the functions \(\arcsin\), \(\arccos\) and \(\arctan\), their relationship to \(\sin\), \(\cos\) and \(\tan\), their graphs and their ranges and domains.
#7.12
Understand and use the definition of a radian and be able to convert between radians and degrees.
#7.13
Know and be able to find the arc length and area of a sector of a circle, when the angle is given in radians.
The results \(s = rθ\) and \(A = \dfrac{1}{2}r^2θ\) where \(θ\) is measured in radians.
#7.14
Understand and use the standard small angle approximations of sine, cosine and tangent.
\(\sin{θ} = θ\), \(\cos{1-\dfrac{θ^2}{2}}\), \(\tan{θ}\) where \(θ\) is in radians.
#7.15
Understand and use the definitions of the \(\sec\), \(\cosec\) and \(\cot\) functions.
Including knowledge of the angles for which they are undefined.
#7.16
Understand relationships between the graphs of the \(\sin\), \(\cos\), \(\tan\), \(\cosec\), \(\sec\) and \(\cot\) functions.
Including domains and ranges.
#7.17
Understand and use the relationships \(\tan^2{θ} + 1 = \sec^2{θ}\) and \(\cot^2{θ} + 1 = \cosec^2{θ}\).
#7.18
Understand and use the identities for \(\sin{(θ \pm ϕ)}\), \(\cos{(θ \pm ϕ)}\), \(\tan{(θ \pm ϕ)}\).
Includes understanding geometric proofs. The starting point for the proof will be given.
[Excludes: Proofs using de Moivre’s theorem will not be accepted.]
#7.19
Know and use identities for \(\sin{2θ}\), \(\cos{2θ}\), \(\tan{2θ}\).
Includes understanding derivations from \(\sin{(θ \pm ϕ)}\), \(\cos{(θ \pm ϕ)}\), \(\tan{(θ \pm ϕ)}\).
\(\cos{2θ} ≡ \cos^2{θ} - \sin^2{θ}\)
\(\cos{2θ} ≡ 2\cos^2{θ} - 1\)
\(\cos{2θ} ≡ 1 - 2\sin^2{θ}\)
#7.20
Understand and use expressions for \(a \cos{θ} \pm b \sin{θ}\) in the equivalent forms \(R\sin{(θ \pm α)}\) and \(R\cos{(θ \pm α)}\).
Includes sketching the graph of the function, finding its maximum and minimum values and solving equations.
#7.21
Use trigonometric identities, relationships and definitions in solving equations.
#7.22
Construct proofs involving trigonometric functions and identities.
#7.23
Use trigonometric functions to solve problems in context, including problems involving vectors, kinematics and forces.
The argument of the trigonometric functions is not restricted to angles.