Game plan 2 · Units 3 and 4

Trigonometry

Non-right-angled triangles, area, bearings and periodic modelling.

Reach for it when

The First Question found no calculus and no statistics. You will be finding sides, angles, area or bearings, or working with a periodic model. Before anything else, check your calculator is in the correct mode: degrees for triangles and bearings, radians for a periodic model.

Part 1

Choosing the rule

Look for an angle–side pair first, which is an angle and the side opposite it. If you have one, you use the sine rule. If you do not, you use the cosine rule.

ABCabc
Each side takes the letter of the angle opposite it. Every rule on this page uses these labels.
Game plan 2.1

Sine rule

  1. 1Identify the angle–side pair, the extra side and the unknown angle. Substitute them into the formula.
  2. 2Rearrange to get $\sin\theta$ on its own.
  3. 3Look at your value of $\sin\theta$. If it is less than $1$, there may be an ambiguous case.
  4. 4Solve for $\theta$ to find the unknown angle.
  5. 5If it is ambiguous, subtract $\theta$ from $180^\circ$ for the second case. Check that the angles work in both triangles.

The sine rule

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
AB2B1Cba
Side $a$ can land in two places, so there are two triangles. The angles at $B_1$ and $B_2$ add to $180^\circ$, which is why step 5 subtracts from $180^\circ$.
Game plan 2.2

Cosine rule

  1. 1Identify the given sides and angles.
  2. 2Confirm that an angle–side pair does not exist.
  3. 3Apply the cosine rule to find the required side or angle.

The cosine rule

$$c^{2}=a^{2}+b^{2}-2ab\cos C$$

Rearranged for an angle (SSS)

$$\cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}$$
Cab?SAS?abcSSS
With SAS you find the third side. With SSS you find an angle, using the rearranged form.

In the Topic Guides

Part 2

Area, bearings and complex problems

Each of these builds on the sine and cosine rules from Part 1.

Game plan 2.3

Area of a triangle

  1. 1Find an angle between two known sides. If there is not one, find unknowns until you have one.
  2. 2Apply the area formula.

Area from two sides and the included angle

$$A=\frac{1}{2}ab\sin C$$
Chab
The height is $h=b\sin C$, so the area is $\frac12\times a\times b\sin C$.
Game plan 2.4

Bearings

  1. 1Draw a diagram if none is given. Mark a north line (a vertical arrow) at each relevant point.
  2. 2Convert each bearing into an angle measured clockwise from the north line at its point.
  3. 3Use parallel line facts to fill in the angles inside the triangle. Co-interior angles add to $180^\circ$ and alternate angles are equal.
  4. 4Once you have an angle–side pair, two sides and the included angle, or all three sides, apply the sine or cosine rule as normal.
  5. 5Give a bearing as three figures measured clockwise from north, for example $072^\circ$T. Use the back bearing ($\pm180^\circ$) to see it from the other point.
NN072°108°AB
The north lines are parallel, so the two marked angles are co-interior and add to $180^\circ$. The bearing of $A$ from $B$ is the back bearing, $072^\circ+180^\circ=252^\circ$T.
Game plan 2.5

Complex modelling

In a complex unfamiliar question, the triangle is usually missing the piece you need, or the trigonometry is only the first half of the question.

  1. 1Draw and fully label a diagram. If none is given, the diagram can earn a mark of its own. Let a variable stand for the unknown, such as $h$, $x$ or $\theta$.
  2. 2Find the triangle that holds the unknown. In 3D, each side may lie on a different face, so use Pythagoras on each face to write the sides in terms of your variable.
  3. 3Look for what you were not told. Parallel lines give alternate angles, and similar triangles give a ratio of sides.
  4. 4Apply the sine rule, cosine rule or area formula, even with the variable still in it. With three sides and no angle, use the cosine rule backwards to find the angle first.
  5. 5Let the last instruction finish it. You might solve for the variable, differentiate to optimise, or integrate for an area.
  6. 6Re-read the question and answer exactly what it asks. A justified decision needs your value compared with the requirement, with units.

In the Topic Guides

Part 3

Common QCAA language: Trigonometry

The wording tells you the context and the rule you need. Look for these phrases.

Note

Trigonometry is different to the other topics. The language tells you the context, but I want you to look just as closely at what you are given in the triangle.

Question typeWhat the question says and gives you
Sine rule“The length of side …”, “the size of angle …”, “how far”, “the distance from …”, often after a bearing or elevation set-up. You are given an angle and the side opposite it, plus one more piece.Your move Know an angle and the side across from it? Reach for the sine rule.
Cosine rule“The length of the side …” (given the two sides either side of an angle), “the size of angle …” (given all three sides). You do not have an angle with its opposite side. You have SAS (find the third side) or SSS (find an angle).Your move Two sides with the angle between them, or all three sides, point you to the cosine rule.
Area of a triangle“The area of the triangle / paddock / region / plot / block”. You need two sides and the angle between them for $\frac12ab\sin C$. Find the angle first if it is missing.Your move Area of a triangle with two sides and the angle between them? Use $\frac12ab\sin C$.
Bearings“Bearing of B from A”, “$052^\circ$T” or “true bearing”, “N40°E”, “due north / south”, “the direction”, and also angle of elevation or depression.Your move Compass directions or the word bearing mean a bearings question. Draw a north line at each point, use the sine or cosine rule, and write the bearing with three figures.
Complex modelling“Make a justified decision”, “does it meet the requirement”, “minimise” or “maximise” with a triangle, “determine the mass” or “the cost”, a 3D object such as a container or tower, parallel walls or fences, or no diagram given.Your move A variable in the triangle, or a final step that asks you to decide, optimise or integrate, makes this complex modelling. Do the trigonometry first, then read the last instruction to see which topic finishes the job.
Periodic modelling“Modelled by $R(t)=a\sin\big(b(t-h)\big)+k$” (or cos), “amplitude”, “period”, “hours after midnight”, tides, temperature, height, Ferris wheel.Your move Something real that repeats, like tides or temperatures, needs a periodic model. Write the sine or cosine model first. The next instruction (solve, differentiate or integrate) tells you what comes after.

In the Topic Guides

To find the equation of a periodic graph, use Game plan 0.3.

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