Trigonometry

The sine and cosine rules, radian measure, the trig identities, and solving trig equations in a given interval.

10 min read

Maths 2 extends trigonometry beyond right-angled triangles to the sine and cosine rules, radians, the key identities, and solving equations across an interval.

For any triangle (sides $a, b, c$ opposite angles $A, B, C$ ):

Sine rule: $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ .
Cosine rule: $a^{2} = b^{2} + c^{2} - 2 b c cos A$ .
Area $= \frac{1}{2} ab sin C$ .

The sine rule has an ambiguous case (angle–side–side): a given pair can correspond to two possible triangles, so check whether an obtuse angle is also valid.

Worked example

A triangle has $b = 7$ , $c = 5$ and $A = 60°$ . Find $a$ .

Cosine rule: $a^{2} = 49 + 25 - 2 (7) (5) cos 60° = 74 - 70 (\frac{1}{2}) = 39$ , so $a = 39 .$

A circle sector of radius r and angle theta, with the arc labelled r theta. — In radians, arc length = rθ and sector area = ½r²θ.

Radians measure angles by arc length: $180° = π$ radians. With $θ$ in radians, a sector of radius $r$ has arc length $r θ$ and area $\frac{1}{2} r^{2} θ$ .

Worked example

A sector has radius $6$ and angle $\frac{π}{3}$ . Find its arc length and area.

Arc $= 6 \times \frac{π}{3} = 2 π$ . Area $= \frac{1}{2} (6)^{2} \cdot \frac{π}{3} = 6 π .$

Two identities you must know: $tan θ = \frac{sin θ}{cos θ}$ and $sin^{2} θ + cos^{2} θ = 1$ .

Know the exact values at $0°, 30°, 45°, 60°, 90°$ (e.g. $sin 30° = \frac{1}{2}$ , $cos 30° = \frac{3}{2}$ , $tan 45° = 1$ ). The sine, cosine and tangent graphs are periodic: $sin$ and $cos$ repeat every $360°$ ( $2 π$ ), $tan$ every $180°$ ( $π$ ).

To solve in a given interval: find the principal value, then use the graph's symmetry/periodicity to find every solution in range. Equations may need an identity first.

Worked example

Solve $12 cos^{2} x + 6 sin x - 10 = 2$ for $0° < x < 360°$ .

Use $cos^{2} x = 1 - sin^{2} x$ : $12 (1 - sin^{2} x) + 6 sin x - 12 = 0 \Rightarrow - 12 sin^{2} x + 6 sin x = 0$ .

$6 sin x (1 - 2 sin x) = 0 \Rightarrow sin x = 0$ or $sin x = \frac{1}{2}$ . In range: $x = 180°, 30°, 150°.$

Exam tip

Always note the interval and use the graph: $sin x = \frac{1}{2}$ has solutions at $30°$ and $150°$ , not just the calculator's first value.

Working in degrees when the sector formulae need radians.
Missing the second (or obtuse) solution of a trig equation.
Forgetting the ambiguous case of the sine rule.

Trigonometry

Overview

The sine & cosine rules

Radian measure

Identities & exact values

Solving trig equations

Common mistakes