Year 10 · Mathematics · Measurement
Right-angled trigonometry
Use the sine, cosine and tangent ratios to find unknown sides and angles in right-angled triangles, including angles of elevation and depression.
Awaiting educator review. This material was drafted by our content team with AI assistance and is waiting for review by a qualified educator. Please check it suits your child before using it. Created 10 October 2026.
Learning goals
Learners will
- Label the hypotenuse, opposite and adjacent sides relative to a given angle.
- Choose the correct trigonometric ratio for a problem.
- Find unknown side lengths and angles in right-angled triangles.
Success looks like
- I can label the sides of a right-angled triangle relative to an angle.
- I can use SOH CAH TOA to pick sine, cosine or tangent.
- I can calculate an unknown side using a calculator in degree mode.
- I can use an inverse trigonometric function to find an angle.
The big idea
Trigonometry connects the angles of a right-angled triangle to the lengths of its sides. First, pick the angle you are working with (often called θ). Then label the sides: the hypotenuse is opposite the right angle, the opposite side is across from θ, and the adjacent side is next to θ.
For a given angle, the ratios between sides are always the same, no matter how big the triangle is. These ratios have names:
- sin θ = opposite ÷ hypotenuse
- cos θ = adjacent ÷ hypotenuse
- tan θ = opposite ÷ adjacent
The memory aid SOH CAH TOA helps: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. Choose the ratio that uses the side you know and the side you want.
Finding a side: write the ratio as an equation and rearrange. If the unknown is on top, multiply; if it is on the bottom, divide. Finding an angle: calculate the ratio, then use the inverse function (sin^−1, cos^−1 or tan^−1) on your calculator.
Make sure the calculator is in degree mode (usually a small D or DEG on the screen). In degree mode, sin 30° = 0.5 exactly. If you get −0.988, the calculator is in radian mode.
Worked examples
Finding the opposite side
A right-angled triangle has a hypotenuse of 10 cm and an angle of 30°. Find the side opposite the 30° angle.
- We know the hypotenuse and want the opposite, so use sine (SOH).
- sin 30° = opposite ÷ 10.
- opposite = 10 × sin 30° = 10 × 0.5 = 5.
Answer: 5 cm
Finding the adjacent side
A right-angled triangle has a hypotenuse of 12 m and an angle of 40°. Find the side adjacent to the 40° angle, correct to one decimal place.
- We know the hypotenuse and want the adjacent, so use cosine (CAH).
- adjacent = 12 × cos 40° ≈ 12 × 0.7660 ≈ 9.19.
- Round to one decimal place.
Answer: 9.2 m
Finding an angle
In a right-angled triangle, the side opposite angle θ is 5 cm and the side adjacent to θ is 8 cm. Find θ, correct to one decimal place.
- We know the opposite and adjacent sides, so use tangent (TOA).
- tan θ = 5 ÷ 8 = 0.625.
- θ = tan^−1(0.625) ≈ 32.005°.
Answer: θ ≈ 32.0°
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which ratio is equal to sin θ?
opposite ÷ hypotenuse — SOH: sine is opposite over hypotenuse.
You know the side opposite an angle and the side adjacent to it. Which ratio links them?
Tangent — TOA: tangent is opposite over adjacent.
A right-angled triangle has a hypotenuse of 20 cm and an angle of 30°. How long is the side opposite the 30° angle?
10 cm — opposite = 20 × sin 30° = 20 × 0.5 = 10 cm. (17.3 cm is the adjacent side, from 20 × cos 30°.)
What is the exact value of tan 45°?
1 — In a right-angled triangle with a 45° angle, the opposite and adjacent sides are equal, so their ratio is 1.
A ramp rises 1 m over a horizontal distance of 4 m. What angle does the ramp make with the ground, correct to one decimal place?
14.0° — tan θ = 1 ÷ 4 = 0.25, so θ = tan^−1(0.25) ≈ 14.04°, which is 14.0° to one decimal place.
What is the exact value of cos 60°? Give your answer as a decimal or fraction.
0.5 — cos 60° = 0.5. You can confirm this on a calculator in degree mode.
A right-angled triangle has a hypotenuse of 15 cm and an angle of 50°. Find the side adjacent to the 50° angle in cm, correct to one decimal place.
9.6 cm — adjacent = 15 × cos 50° ≈ 15 × 0.6428 ≈ 9.64, which is 9.6 cm to one decimal place.
In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 14 cm. Find θ in degrees.
30° — sin θ = 7 ÷ 14 = 0.5, so θ = sin^−1(0.5) = 30°.
Watch out for
- Labelling the opposite and adjacent sides without reference to the chosen angle. These labels change if you use the other acute angle; only the hypotenuse stays the same.
- Calculator in radian mode, which gives answers that look strange. Check that sin 30° gives 0.5 before starting.
- Dividing when you should multiply. Write the ratio as an equation first, then rearrange step by step.
- Using sin instead of sin^−1 when finding an angle. If the unknown is an angle, you need an inverse function.
Tips for parents
- Ask your learner to mark the angle, then label H, O and A on every diagram before calculating anything.
- Encourage a reasonableness check: a side opposite a small angle should be short, and no side can be longer than the hypotenuse.
- Make up a silly sentence together to remember SOH CAH TOA. Learners remember the ones they invent.
Go further
- Estimate the height of a tree or building: stand a measured distance away, measure the angle of elevation with a protractor and a straw, and use tan. Remember to add your eye height.
- Use a calculator to explore sin θ and cos θ for angles from 0° to 90°. What patterns do you notice? When is sin θ = cos θ?
- Investigate how surveyors or navigators use trigonometry to find distances they cannot measure directly.