QCE Mathematical Methods - Unit 4 - Trigonometry

Sine Rule and Cosine Rule | QCE Mathematical Methods

Learn QCE Mathematical Methods sine rule, cosine rule, area of a triangle, ambiguous case and non-right-triangle modelling.

Updated 2026-05-18 - 4 min read

The sine and cosine rules extend trigonometry beyond right-angled triangles. They are especially useful in bearings, angles of elevation, distances and two- or three-dimensional contexts.

Unit circle and exact values

The unit circle gives the foundation for sine, cosine and tangent. On a unit circle, the radius is $1$, so the point at angle $\theta$ has coordinates:

$ (\cos\theta,\sin\theta). $

That is why cosine is the horizontal coordinate and sine is the vertical coordinate. The special triangles give exact values:

| angle | $\sin\theta$ | $\cos\theta$ | $\tan\theta$ | | --- | --- | --- | --- | | $30^\circ$ or $\frac{\pi}{6}$ | $\frac12$ | $\frac{\sqrt3}{2}$ | $\frac1{\sqrt3}$ | | $45^\circ$ or $\frac{\pi}{4}$ | $\frac{\sqrt2}{2}$ | $\frac{\sqrt2}{2}$ | $1$ | | $60^\circ$ or $\frac{\pi}{3}$ | $\frac{\sqrt3}{2}$ | $\frac12$ | $\sqrt3$ |

Radians and degrees both appear in Methods. To convert:

$ 180^\circ=\pi\text{ radians}. $

Use radians for calculus and use whichever unit the trigonometry context asks for.

Transformed trig models

For:

$ y=a\sin(n(x+b))+c $

or:

$ y=a\cos(n(x+b))+c, $

the amplitude is $|a|$, the midline is $y=c$, and the period is:

$ \frac{2\pi}{|n|}. $

For tangent:

$ y=a\tan(n(x+b))+c, $

the period is:

$ \frac{\pi}{|n|}. $

Tangent also has vertical asymptotes where the inside angle makes $\tan$ undefined, such as $\frac{\pi}{2}+k\pi$ for the basic tangent graph.

Trig modelling questions often ask you to connect these parameters to context. If tide height is modelled by:

$ h(t)=7\cos\left(\frac{\pi t}{12}\right)+12, $

then the high tide is $12+7=19$, the low tide is $12-7=5$, and the period is:

$ \frac{2\pi}{\pi/12}=24 $

hours. To find when the height is $6$, solve the trig equation and keep all solutions in the requested time interval.

Sine rule

The sine rule links each side to its opposite angle:

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

Use it when you have a matched side-angle pair and need another side or angle.

A quick rule choice guide:

| given information | usually use | | --- | --- | | one side-angle opposite pair and another side or angle | sine rule | | two sides and included angle | cosine rule | | three sides | cosine rule | | two sides and included angle for area | $\frac12bc\sin A$ |

Cosine rule

The cosine rule is:

$ c^2=a^2+b^2-2ab\cos(C) $

Use it when you have two sides and the included angle, or all three sides and need an angle.

Area formula

For two sides and the included angle:

$ \text{Area}=\frac{1}{2}bc\sin(A) $

This is useful when the triangle is not right-angled and height is not directly given.

Ambiguous sine rule

The ambiguous case can occur when you are given two sides and a non-included angle. Since $\sin(\theta)=\sin(180^\circ-\theta)$ for angles in a triangle context, there may be two possible triangles.

This is not a calculator error. It is geometry. If the context is bearings or a physical diagram, one of the triangles may be impossible.

For example, if:

$ \sin\theta=\frac{9\sin22^\circ}{5}, $

technology may give $\theta\approx42.4^\circ$. The second possible triangle uses:

$ 180^\circ-42.4^\circ=137.6^\circ. $

Both are possible only if the triangle angle sum and the context allow them. If another given angle is $70^\circ$, then $137.6^\circ$ would already push the total above $180^\circ$, so it must be rejected.

Worked example

Modelling contexts

In applications, spend time drawing the triangle:

• mark north lines for bearings
• label angles of elevation or depression from the horizontal
• convert compass information into interior triangle angles
• check whether a 3D problem can be split into right triangles or non-right triangles

Bearings are measured clockwise from north. In a multi-leg journey, draw a north line at each point. Parallel north lines create alternate angles, which often lets you find the interior angle of the triangle before using sine or cosine rule.

In 3D problems, draw the full diagram, then redraw the relevant 2D triangle separately. Most 3D Methods problems are solved by breaking the shape into one triangle at a time rather than trying to stare at the whole diagram.

Quick check

Sources

• QCAA Mathematical Methods subject page
• QCAA Mathematical Methods 2025 syllabus
• Monash Student Academic Success: Sine and Cosine Rules
• OpenStax Algebra and Trigonometry 2e: Non-right Triangles - Law of Sines