Find a numerator side
A ratio connects a known length to the required leg. Multiply both sides of the equation by the known denominator
- Identify the pair Name both sides from the marked angle. The unknown must be the numerator in the chosen ratio \(\displaystyle \begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\)
- Multiply both sides Multiply by the known nonzero denominator; the same operation on both sides preserves equality \(\displaystyle \begin{aligned}\sin\theta&=\frac{x}{h}\\h\sin\theta&=x\end{aligned}\)
- Evaluate and check Use degree mode and keep full precision until the final length. A sine or cosine result is shorter than the hypotenuse. A tangent result may exceed the known adjacent leg
- Combine two right triangles For a perpendicular altitude with its foot between two outer base points, calculate the two base segments separately and add them. Read whether each angle is from the base or altitude \(\displaystyle AC=AB+BC\)
Worked example
A right triangle has angle 37°, hypotenuse 18 cm and opposite side x. Find x to one decimal place
\[\sin37^{\circ}=\frac{x}{18}\]
- \(\displaystyle x=18\sin37^{\circ}\)Multiply both sides by the known hypotenuse
- \(\displaystyle x\approx10.832670\text{ cm}\)Use degree mode and retain the full calculator value
- \(\displaystyle x\approx10.8\text{ cm}\)Round once; the leg is shorter than the hypotenuse
Common mistakes
Dividing when the unknown is on top
✕\(\displaystyle \begin{gathered}\sin37^{\circ}=\frac{x}{18}\\x=\frac{18}{\sin37^{\circ}}\approx29.9\text{ cm}\end{gathered}\)✓\(\displaystyle \begin{gathered}\sin37^{\circ}=\frac{x}{18}\\x=18\sin37^{\circ}\approx10.8\text{ cm}\end{gathered}\)
Multiply both sides by the known denominator. Dividing would incorrectly make this leg longer than the hypotenuseRounding too soon
✕\(\displaystyle 18\times0.60\)✓\(\displaystyle 18\sin37^{\circ}\)
Keep the full degree-mode ratio until the requested final roundingDifficulty levels
introFind a numerator side with sine, cosine or tangent to one decimal place
standardUse general angles and report the numerator length to two decimal places
challengeAdd two base segments using a shared perpendicular altitude
examDistinguish correct numerator lengths from wrong-operation and wrong-ratio choices
Questions about this topic
Why do I multiply?+
The unknown is divided by the known denominator. Multiplying both sides by that denominator leaves the unknown alone
Can the answer exceed the given length?+
Not when the given length is the hypotenuse. It can when tangent links the two legs
Do I add or subtract two base parts?+
Read the endpoints. If the altitude foot is between the outer endpoints, the full base is their sum
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry
In this chapter
9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions