Trigonometry

Pythagoras' theorem

$3^2 + 3^2 = c^2 \quad\Rightarrow\quad c = \sqrt{18} = 4.2\text{ cm}$

Right-angled triangles

For an angle $\theta$ in a right-angled triangle: the hypotenuse is the longest side, the opposite side is the side facing $\theta$, and the adjacent side is the side next to $\theta$ (that is not the hypotenuse).

Since we have the lengths of the opposite side and the hypotenuse, we use sin:

$\sin(\theta) = \frac{3}{5} \quad\Rightarrow\quad \theta = \sin^{-1}\left(\frac{3}{5}\right) = 36.9^\circ$

The shortest distance from a point to a line is the perpendicular distance.

Since the shortest distance is the perpendicular distance, we calculate the side opposite the angle using sine:

$\sin(30^\circ) = \frac{x}{5} \quad\Rightarrow\quad x = 5\sin(30^\circ) = 2.5\text{ cm}$

$\tan(\theta) = \frac{1}{4} \quad\Rightarrow\quad \theta = \tan^{-1}\left(\frac{1}{4}\right) = 14.0^\circ$

Write the answer to 1 decimal place even if it is $.0$.

Non-right-angled triangles

Sine rule

The 4 cm side is opposite the $40^\circ$ angle, and the 2 cm side is opposite $A$:

$\frac{4}{\sin 40^\circ} = \frac{2}{\sin A} \quad\Rightarrow\quad \sin A = \frac{2\sin 40^\circ}{4} \quad\Rightarrow\quad A = 18.7^\circ$

Cosine rule

Use the cosine rule when you know:

• all three sides (SSS), or
• two sides and the angle between them (SAS).

$A^2 = 5^2 + 3^2 - 2(5)(3)\cos(80^\circ) = 34 - 30\cos 80^\circ$

$A = 5.37\text{ cm}$

Area rule

Use it when you know two sides and the angle between them (SAS).

$\text{Area} = \tfrac{1}{2}(5)(3)\sin 80^\circ = 7.4\ \text{cm}^2$

Exact trigonometric values

You must know these values without a calculator:

Trigonometric functions

The graphs, for $0^\circ \le x \le 360^\circ$

• $y = \sin x$ starts at 0, rises to a maximum of 1 at $90^\circ$, is 0 at $180^\circ$, falls to $-1$ at $270^\circ$ and is 0 at $360^\circ$.
• $y = \cos x$ starts at 1, is 0 at $90^\circ$, is $-1$ at $180^\circ$, 0 at $270^\circ$ and 1 at $360^\circ$. It is the sine graph shifted $90^\circ$ to the left.
• $y = \tan x$ is 0 at $0^\circ$, $180^\circ$ and $360^\circ$, rises steeply towards $90^\circ$ and $270^\circ$, where it is undefined (the graph has asymptotes).

Sine and cosine repeat every $360^\circ$ and stay between $-1$ and $1$. Tangent repeats every $180^\circ$.

Solving trigonometric equations

A calculator gives one answer. Use the graph's symmetry to find the other one in $0^\circ \le x \le 360^\circ$.