Need to Know

Trigonometry

The rules you must have ready for the exam.

How to use this: say each rule in your head, then tap Show me to check.

Right-angled triangles

6 rules

Rule 01

Label the sides.

Hypotenuse = longest, opposite the right angle. Opposite faces your angle. Adjacent is beside it.

Rule 02

SOH CAH TOA.

$\sin A = \tfrac{\text{opp}}{\text{hyp}}$, $\cos A = \tfrac{\text{adj}}{\text{hyp}}$, $\tan A = \tfrac{\text{opp}}{\text{adj}}$.

Rule 03

Choosing the ratio.

Label the side you have and the side you want; pick the ratio using both.

Rule 04

x on the bottom.

If the unknown is underneath, divide: $x = \dfrac{\text{side}}{\text{ratio of the angle}}$.

Rule 05

Finding an angle.

Know two sides → work out the ratio, then press $\sin^{-1}$, $\cos^{-1}$ or $\tan^{-1}$.

Rule 06

Pythagoras.

For the third side: $H^2 = O^2 + A^2$.

2 rules

Rule 07

Angle of elevation.

The angle you look up from the horizontal.

Rule 08

Angle of depression.

The angle you look down from the horizontal; equal to the elevation looking back up.

4 rules

Rule 09

Label opposite sides.

Side $a$ faces angle $A$, side $b$ faces $B$, side $c$ faces $C$.

Rule 10

Sine Rule.

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B}$. Use with an angle and its opposite side plus one more. Flip it ($\tfrac{\sin A}{a}$) to find an angle.

Rule 11

Cosine Rule.

$a^2 = b^2 + c^2 - 2bc\cos A$. Use for two sides and the angle between, or for three sides (rearrange for $\cos A$; a negative answer means obtuse).

Rule 12

Area of a triangle.

$\text{Area} = \tfrac{1}{2}ab\sin C$ — two sides and the angle between. Answer in square units.

Mathslive.ie · Trigonometry · Need to Know