Finding an unknown side
Name the given and unknown side roles before choosing a ratio. After substitution, multiply or divide according to whether the unknown is in the numerator or denominator
\[\sin\theta=\frac{o}{h},\quad\cos\theta=\frac{a}{h},\quad\tan\theta=\frac{o}{a}\]
- Name both side roles Identify the given side and the unknown side as opposite, adjacent or hypotenuse relative to θ \(\displaystyle \theta=60^{\circ},\quad o=12,\quad h=x\)
- Use the matching ratio Choose the ratio whose numerator and denominator contain the two side roles \(\displaystyle \sin60^{\circ}=\frac{12}{x}\)
- Multiply for an unknown numerator If the unknown is on top, multiply the known denominator by the trigonometric value \(\displaystyle \cos40^{\circ}=\frac{x}{12}\Rightarrow x=12\cos40^{\circ}\)
- Divide for an unknown denominator If the unknown is underneath, multiply by it first and divide the known length by the trigonometric value \(\displaystyle \sin60^{\circ}=\frac{12}{x}\Rightarrow x=\frac{12}{\sin60^{\circ}}\)
Worked example
A right-angled triangle has θ = 37°, hypotenuse 18 cm and opposite side x. Find x to 1 decimal place
\[\sin37^{\circ}=\frac{x}{18}\]
- \(\displaystyle \sin37^{\circ}=\frac{x}{18}\)The unknown is opposite and the given side is the hypotenuse, so sine is the matching ratio
- \(\displaystyle x=18\sin37^{\circ}\)Multiply both sides by 18 because the unknown is in the numerator
- \(\displaystyle x=18\sin37^{\circ}\approx10.8327\ \mathrm{cm}\)Evaluate the trigonometric value in degree mode without rounding the intermediate product
- \(\displaystyle x\approx10.8\ \mathrm{cm}\)Round the side length to 1 decimal place and include the unit
Common mistakes
Multiplying when the unknown is underneath
✕\(\displaystyle \sin60^{\circ}=\frac{12}{x}\Rightarrow x=12\times0.8660\approx10.4\text{ cm}\)✓\(\displaystyle \sin60^{\circ}=\frac{12}{x}\Rightarrow x\approx\frac{12}{0.8660}\approx13.9\text{ cm}\)
An unknown denominator is isolated by dividing the known length by the trigonometric valueDividing when the unknown is on top
✕\(\displaystyle \sin37^{\circ}=\frac{x}{18}\Rightarrow x=\frac{18}{\sin37^{\circ}}\approx29.9\text{ cm}\)✓\(\displaystyle \sin37^{\circ}=\frac{x}{18}\Rightarrow x=18\sin37^{\circ}\approx10.8\text{ cm}\)
An unknown numerator is isolated by multiplying the known denominator by the trigonometric valueChoosing the wrong ratio
✕\(\displaystyle \cos60^{\circ}=\frac{12}{x}\)✓\(\displaystyle \sin60^{\circ}=\frac{12}{x}\)
The opposite side and hypotenuse form the sine ratio for the marked angleDifficulty levels
introFind a numerator side using sine, cosine or tangent and round to one decimal place
standardFind a numerator or denominator side using sine, cosine or tangent and round to two decimal places
challengeFind two base segments from a shared perpendicular altitude and two marked angles, then add their unrounded lengths and round to two decimal places
examChoose the side length from direction and wrong-ratio distractors
Questions about this topic
How do I know whether to multiply or divide?+
After writing the ratio, check the position of the unknown. An unknown numerator is the known denominator multiplied by the ratio value; an unknown denominator is the known numerator divided by the ratio value
Why should an unknown hypotenuse be longer than the known leg?+
The hypotenuse is the longest side of a right-angled triangle. If a calculation gives a shorter hypotenuse, the ratio or rearrangement is wrong
Why do I need to name the side roles first?+
Sine, cosine and tangent use different pairs of side roles. Naming the roles prevents a ratio from being chosen from the wrong sides
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 11: Trigonometry
In this chapter
Name sides and state trigonometric ratios11A Trigonometric ratios: Evaluate a trigonometric ratio, and find an angle from its valueChoose a trigonometric ratioFind an unknown side of a right-angled triangle11C Finding unknown angles: Evaluate a trigonometric ratio, and find an angle from its valueFind an angle of a right-angled triangle from two sidesSolve a practical right-angled-triangle problemAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsUse the sine rule in a non-right-angled triangleThe ambiguous case of the sine ruleUse the cosine rule in a non-right-angled triangle11H Cosine rule: Choose the rule a triangle problem needs11H Cosine rule: Solve a practical triangle problem end to end11I Triangle area: Choose the rule a triangle problem needsFind the area of a triangle11I Triangle area: Solve a practical triangle problem end to end