Year 9 - Trigonometry - Free practice

Year 9 Trigonometry Worksheets & Practice Tests

SOHCAHTOA - find missing sides and angles in right-angled triangles.

Quick answer

How do you use trigonometry to find missing sides and angles in right-angled triangles?

Label the triangle's sides as opposite, adjacent, or hypotenuse relative to the known angle, then choose sin, cos, or tan: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Rearrange to find a missing side, or use an inverse ratio (e.g. sin⁻¹) to find a missing angle. For example, if the opposite side is 5 and hypotenuse is 10, the angle is sin⁻¹(0.5) = 30°.

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Worked examples

Year 9 trigonometry: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Labelling sides and choosing a ratio
A right-angled triangle has an acute angle of $35°$ . The hypotenuse is $12$ cm. Find the side opposite the $35°$ angle, correct to two decimal places.
Show worked solution
Key idea: Identify the sides relative to the given angle, then choose the ratio that links those two sides.
1. The known side is the hypotenuse ( $12$ cm). The unknown side is opposite the $35°$ angle. Use sine: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$
2. Substitute the known values: $\sin 35° = \frac{x}{12}$
3. Solve for $x$ : $x = 12 \times \sin 35°$
4. Calculate: $x = 12 \times 0.5736 \approx 6.88 \text{ cm}$
Answer: The opposite side is approximately $6.88$ cm.
Example 2 · Finding a missing side using tangent
In a right-angled triangle, an acute angle measures $52°$ . The side adjacent to that angle is $9$ m. Find the length of the side opposite the $52°$ angle, correct to two decimal places.
Show worked solution
Key idea: When the opposite and adjacent sides are involved, use tangent.
1. Write the tangent ratio: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$
2. Substitute the known values: $\tan 52° = \frac{x}{9}$
3. Solve for $x$ : $x = 9 \times \tan 52°$
4. Calculate: $x = 9 \times 1.2799 \approx 11.52 \text{ m}$
Answer: The opposite side is approximately $11.52$ m.
Example 3 · Finding the angle of a ramp
A loading ramp rises $1.4$ m vertically over a horizontal distance of $5$ m. Find the angle the ramp makes with the horizontal, correct to the nearest degree.
Show worked solution
Key idea: The vertical rise is opposite the unknown angle and the horizontal distance is adjacent to it, so use inverse tangent.
1. Write the tangent ratio: $\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{1.4}{5}$
2. Simplify: $\tan\theta = 0.28$
3. Apply inverse tangent: $\theta = \tan^{-1}(0.28)$
4. Calculate: $\theta \approx 15.64° \approx 16°$
Answer: The ramp makes an angle of approximately $16°$ with the horizontal.

Common mistakes

Where students usually lose marks

Labelling opposite and adjacent from the wrong angle

Using the side next to the right angle as the opposite instead of checking which angle is $\theta$ .

Always label opposite, adjacent, and hypotenuse relative to the angle you are working with, not the right angle. The opposite side is the one that does not touch $\theta$ at all.

Choosing the wrong trigonometric ratio

Writing $\sin\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ when the opposite side is known.

Identify which two sides are involved (opposite, adjacent, or hypotenuse), then match them to SOH-CAH-TOA. Only pick your ratio after labelling the sides.

Forgetting to use inverse trig when finding an angle

Writing $\tan\theta = 0.75$ , then stating $\theta = 0.75°$ .

When the ratio is known and the angle is unknown, apply the inverse function: $\theta = \tan^{-1}(0.75)$ . The ratio and the angle are different things.

Dividing instead of multiplying when solving for a missing side

From $\sin 40° = \frac{x}{15}$ , writing $x = 15 \div \sin 40°$ instead of $x = 15 \times \sin 40°$ .

Rearrange the equation carefully before reaching for the calculator. Multiply both sides by the hypotenuse to isolate the opposite side: $x = 15 \times \sin 40°$ .

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Your turn

A ramp rises $1.8$ m vertically over a horizontal distance of $4.5$ m. Find the angle the ramp makes with the horizontal, correct to 2 decimal places. Give your answer in degrees.

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About this worksheet

Why we built it

Year 9 trig is the first proper introduction to SOHCAHTOA. These worksheets give your student practice on both directions - finding a missing side, finding a missing angle - across enough varied triangles to make the right-ratio choice feel automatic.

What's covered

Sub-skills your student will practise

✓Labelling opposite / adjacent / hypotenuse for a given angle
✓Choosing the right ratio (sin / cos / tan) for the question
✓Finding a missing side length
✓Finding a missing angle using inverse trig
✓Word problems involving heights, ramps, shadows

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Frequently asked

Questions parents ask about trigonometry

What does SOHCAHTOA stand for?

Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. It's a memory trick for the three trig ratios in a right-angled triangle.

How do students choose which ratio to use?

Look at the two sides involved in the question relative to the given angle. If it's opposite and hypotenuse, use sine. Adjacent and hypotenuse, cosine. Opposite and adjacent, tangent.

How do students find a missing angle (not a side)?

Set up the trig ratio with the known sides, then apply the inverse function (sin^-1, cos^-1, tan^-1). For sin theta = 0.5, theta = sin^-1(0.5) = 30 degrees.

Why does Year 9 trigonometry only cover right-angled triangles?

Because non-right-angle trig needs the sine and cosine rules, which are Year 10A and beyond. The SOHCAHTOA toolkit covers right-angled triangles fully but doesn't extend to other shapes.

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