Sides, ratios and similar triangles
Start with the right angle, then the marked acute angle. Ratios describe the triangle's shape, so enlarging it does not change them
\[\begin{aligned}\sin\theta&=\frac{o}{h}\\\cos\theta&=\frac{a}{h}\\\tan\theta&=\frac{o}{a}\end{aligned}\]
- Hypotenuse The side opposite the right angle is the hypotenuse. Changing the marked acute angle does not change this side
- Opposite and adjacent The opposite leg is across from the marked angle. The adjacent leg touches it. Switching acute angles exchanges these two roles
- Similar triangles Multiplying every side by the same positive factor preserves the angles and every ratio of corresponding lengths \(\displaystyle \begin{aligned}\frac{ko}{kh}&=\frac{o}{h}\\\frac{ka}{kh}&=\frac{a}{h}\\\frac{ko}{ka}&=\frac{o}{a}\qquad(k>0)\end{aligned}\)
- Ratios have no length unit Use the same unit for both side lengths. Their units cancel when you divide
- Check the size For an acute angle in a right triangle, sine and cosine are positive and less than one. Tangent may be less than, equal to or greater than one
- Complementary acute angles The acute angles in a right triangle sum to 90°. Exchanging them exchanges the legs, so the sine of one equals the cosine of the other \(\displaystyle \begin{gathered}\alpha+\beta=90^{\circ}\\\sin\alpha=\cos\beta\end{gathered}\)
Worked example
Triangle PQR is right-angled at R and the marked angle theta is at P. Use the supplied lengths to write its cosine as an exact fraction
\[\begin{aligned}PQ&=82\text{ cm}\\PR&=80\text{ cm}\\QR&=18\text{ cm}\end{aligned}\]
- \(\displaystyle h=PQ\)This side lies opposite the right angle
- \(\displaystyle a=PR\)This leg touches the marked angle
- \(\displaystyle \cos\theta=\frac{a}{h}\)Cosine uses the adjacent leg and the hypotenuse
- \(\displaystyle \cos\theta=\frac{80}{82}\)Substitute the supplied lengths
- \(\displaystyle \cos\theta=\frac{40}{41}\)Divide numerator and denominator by two. Either exact fraction has the same value
Common mistakes
Calling the hypotenuse adjacent
✕\(\displaystyle a=MN\)✓\(\displaystyle a=MO\)
In triangle MNO, the right angle is at O and the marked acute angle is at M. MN is the hypotenuse; MO is the adjacent leg. Here a denotes the adjacent length, so it is the length MO, not MNUsing the other leg for cosine
✕\(\displaystyle \cos\theta=\frac{o}{h}\)✓\(\displaystyle \cos\theta=\frac{a}{h}\)
The opposite side belongs in the sine ratio. Cosine uses the adjacent sideReversing tangent
✕\(\displaystyle \tan\theta=\frac{a}{o}\)✓\(\displaystyle \tan\theta=\frac{o}{a}\)
Tangent divides the opposite length by the adjacent length in that orderDifficulty levels
introRecognise a side and write one ratio. Practise sine, cosine and tangent in turn
standardWrite all three ratios from a labelled triangle, with either acute angle marked
challengeWrite all six ratios for two similar right triangles drawn in different orientations
examRecognise an exact sine, cosine or tangent ratio from a diagram
Questions about this topic
Does the hypotenuse change when the marked angle changes?+
No. The right angle fixes the hypotenuse. Only the opposite and adjacent legs exchange roles
Why can triangles of different sizes have the same ratio?+
When their corresponding angles agree, their side lengths share a scale factor. That factor cancels from the numerator and denominator
Can a tangent ratio be greater than one?+
Yes. That happens when the opposite leg is longer than the adjacent leg
Must the two lengths use the same unit?+
Yes. Convert 250 mm to 25 cm before dividing by 65 cm. Dividing 250 by 65 without conversion gives the wrong ratio. Ratios of lengths then have no unit
Can I use a calculator?+
No. This topic is practised without a 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