Coordinate Geometry and Trigonometry

Distance, midpoint, and gradient

Distance between $(x_1,y_1)$ and $(x_2,y_2)$: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$.

Midpoint: $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Gradient: $m=\frac{y_2-y_1}{x_2-x_1}$.

Equation of a line

The equation of a line through $(x_1,y_1)$ with gradient $m$ is $y-y_1=m(x-x_1)$. Parallel lines have equal gradients; perpendicular lines have gradients that multiply to $-1$.

Trigonometric ratios

In a right triangle, for a chosen angle: $\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$. Remember this as SOHCAHTOA.

Sine and cosine rule

For any triangle (not just right-angled): sine rule $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$; cosine rule $a^2=b^2+c^2-2bc\cos A$.

Angles of elevation, depression, and bearings

Angle of elevation is measured upward from the horizontal; angle of depression downward. Bearings are always measured clockwise from North as a three-digit angle, e.g. due East is $090°$ and due South-West is $225°$.