Trigonometry

Identities

$$\sin^2\theta + \cos^2\theta = 1, \qquad \tan\theta = \frac{\sin\theta}{\cos\theta}$$

	$\sin$	$\cos$	$\tan$
$90° - \theta$	$\cos\theta$	$\sin\theta$	$\dfrac{1}{\tan\theta}$
$180° - \theta$	$\sin\theta$	$-\cos\theta$	$-\tan\theta$
$180° + \theta$	$-\sin\theta$	$-\cos\theta$	$\tan\theta$
$360° - \theta$	$-\sin\theta$	$\cos\theta$	$-\tan\theta$

Solving trigonometric equations ($0° \leq \theta < 360°$)

1. Find the basic angle from the calculator.
2. Use the sign of the ratio to find which quadrants the answers are in ("All Students Take Chemistry": all positive in quadrant I, sin in II, tan in III, cos in IV).

Example: $2\sin\theta = 1 \Rightarrow \sin\theta = \frac{1}{2}$. The basic angle is $30°$ and sin is positive in quadrants I and II, so $\theta = 30°$ or $150°$.

Common mistake: stopping at one answer. Most trig equations have two solutions in $0°$ to $360°$.

Any triangle $ABC$

Side $a$ is opposite angle $A$, and so on.

Sine rule (two angles and a side, or two sides and a non-included angle):

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Cosine rule (two sides and the included angle, or three sides):

$$c^2 = a^2 + b^2 - 2ab\cos C, \qquad \cos C = \frac{a^2 + b^2 - c^2}{2ab}$$

Area:

$$\text{Area} = \frac{1}{2}ab\sin C \qquad \text{or Heron's formula} \quad \sqrt{s(s - a)(s - b)(s - c)}, \; s = \frac{a + b + c}{2}$$

Exam tip: in 3D problems, draw the right-angled triangle you are using separately. The angle between a line and a plane is the angle between the line and its projection on the plane.