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Trigonometry
O-Level A Math
Special Ratios
$30^\circ$$45^\circ$$60^\circ$
$\sin$$\dfrac{1}{2}$$\dfrac{1}{\sqrt{2}}$$\dfrac{\sqrt{3}}{2}$
$\cos$$\dfrac{\sqrt{3}}{2}$$\dfrac{1}{\sqrt{2}}$$\dfrac{1}{2}$
$\tan$$\dfrac{1}{\sqrt{3}}$$1$$\sqrt{3}$
Pythagorean Identities
$$\sin^2\theta + \cos^2\theta = 1$$
$$1 + \tan^2\theta = \sec^2\theta$$
$$1 + \cot^2\theta = \csc^2\theta$$
$$\cos^2\theta = 1 - \sin^2\theta$$
Double Angle Formulae
$$\sin 2A = 2\sin A \cos A$$
$$\cos 2A = \cos^2 A - \sin^2 A$$
$$\cos 2A = 2\cos^2 A - 1 \implies \cos^2 A = \dfrac{1 + \cos 2A}{2}$$
$$\cos 2A = 1 - 2\sin^2 A \implies \sin^2 A = \dfrac{1 - \cos 2A}{2}$$
$$\tan 2A = \dfrac{2\tan A}{1 - \tan^2 A}$$
Other Ways To Use Double Angle Formula
$$\sin 4\theta = 2\sin 2\theta\,\cos 2\theta$$
$4\theta = 2 \times (2\theta)$, angle is doubled
Half-Angle Formulae
$$\sin\theta = 2\sin\dfrac{\theta}{2}\cos\dfrac{\theta}{2}$$
$$\begin{aligned} \cos A &= \cos^2\!\dfrac{A}{2} - \sin^2\!\dfrac{A}{2} \\[6pt] &= 2\cos^2\!\dfrac{A}{2} - 1 \\[6pt] &= 1 - 2\sin^2\!\dfrac{A}{2} \end{aligned}$$
$$\tan A = \dfrac{2\tan\dfrac{A}{2}}{1 - \tan^2\dfrac{A}{2}}$$
Addition Formulae
$$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$$
$$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$$
$$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$$
Can be applied to find e.g. $\sin 3x = \sin(2x + x)$
R-Formula
$$a\sin\theta \pm b\cos\theta = R\sin(\theta \pm \alpha)$$
$$a\cos\theta \pm b\sin\theta = R\cos(\theta \mp \alpha)$$
where $R = \sqrt{a^2 + b^2}$ and $\tan\alpha = \dfrac{b}{a}$, with $a,\,b > 0$
Sine Graph
yx0π2π3π22π1−1y = sin x
Amplitude: 1   Period: 360° or 2π
Principal range: −90° ≤ x ≤ 90°
Cosine Graph
yx0π2π3π22π1−1y = cos x
Amplitude: 1   Period: 360° or 2π
Principal range: 0° ≤ x ≤ 180°
Tangent Graph
yx0π2π3π22πy = tan x
Period: 180° or π   (no amplitude)
Principal range: −90° < x < 90°
Drawing Trigonometric Graphs
$$y = a \sin bx + c$$
$$y = a \cos bx + c$$
$$y = a \tan bx + c$$
a = amplitude
b = no. of cycles in 360° or 2π
c = no. of units to shift the graph up/down

Formulas for the Singapore O-Level Additional Mathematics syllabus