Grasp Maths

Year 10

Trigonometry in right-angled triangles

Use the sine, cosine and tangent ratios (sin⁡θ=OH\sin\theta = \frac{O}{H}, cos⁡θ=AH\cos\theta = \frac{A}{H}, tan⁡θ=OA\tan\theta = \frac{O}{A}) to find missing sides and angles in right-angled triangles.

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Lesson overview

Geometry - trigonometric ratios

Trigonometry is the mathematics of triangles. In a right-angled triangle, we can use three basic ratios to relate the angles and sides. For any angle θ\theta (other than the right angle), we label: OO = the side opposite to the angle, AA = the side adjacent (next to) the angle, and HH = the hypotenuse (the longest side, opposite the right angle). The three ratios are: sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, and tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}. A common mnemonic is SOHCAHTOA. Using these ratios, we can find missing sides if we know an angle and a side, or find missing angles if we know two sides.

Finding a missing side using sine

In a right-angled triangle, the hypotenuse is 10 cm and one of the other angles is 35°. Find the opposite side.

?10 cm35°

We know the hypotenuse and want the opposite side, so use sine. Multiply both sides by 10.

Finding a missing side using cosine

In a right-angled triangle, the hypotenuse is 8 m and one angle is 50°. Find the adjacent side.

?8 m50°

We know the hypotenuse and want the adjacent side, so use cosine.

Finding a missing side using tangent

In a right-angled triangle, the adjacent side is 6 cm and the angle is 40°. Find the opposite side.

?6 cm40°

We know the adjacent side and want the opposite side, so use tangent. Neither involves the hypotenuse.

Finding a missing angle using inverse trigonometry

In a right-angled triangle, the opposite side is 5 cm and the hypotenuse is 12 cm. Find the angle.

5 cm12 cm?

Use the inverse (or reciprocal) function: sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1}, or tan⁡−1\tan^{-1} to find the angle.

Worked example

A ladder leans against a wall. The ladder is 5 m long and makes an angle of 65° with the ground. How high up the wall does the ladder reach?

? (height)5 m (ladder)65°
  1. Draw a diagram. The ladder is the hypotenuse (5 m), the height on the wall is the opposite side, and the angle with the ground is 65°.
  2. Identify which ratio to use. We know the hypotenuse and want to find the opposite side, so we use sine.
  3. Write the ratio: sin⁡(65°)=height5\sin(65°) = \frac{\text{height}}{5}.
  4. Rearrange to find the height: height =5×sin⁡(65°)= 5 \times \sin(65°).
  5. Use a calculator: sin⁡(65°)≈0.906\sin(65°) ≈ 0.906.
  6. Calculate: height =5×0.906≈4.53= 5 \times 0.906 ≈ 4.53 m.
  7. The ladder reaches approximately 4.53 m up the wall.

Try it

For each triangle, identify which sides or angles are known and which you need to find, then choose the correct trigonometric ratio (sine, cosine, or tangent).

Question 1

In a right-angled triangle, the hypotenuse is 10 cm and one angle is 30°. What is the length of the side opposite to the 30° angle? (Use sin⁡(30°)=0.5\sin(30°) = 0.5.)

?10 cm30°

💡 Use sin⁡(30°)=opposite10\sin(30°) = \frac{\text{opposite}}{10}. So opposite =10×sin⁡(30°)=10×0.5=5= 10 \times \sin(30°) = 10 \times 0.5 = 5 cm.

Question 2

In a right-angled triangle, the hypotenuse is 13 cm and the adjacent side is 5 cm. Which trigonometric ratio would you use to find the angle?

5 cm13 cm?

💡 You know the adjacent side and the hypotenuse, so use cosine: cos⁡(θ)=513\cos(\theta) = \frac{5}{13}. Thus θ=cos⁡−1(5/13)\theta = \cos^{-1}(5/13).

Question 3

In a right-angled triangle, the adjacent side is 7 cm and the angle is 40°. What is the length of the opposite side? (Use tan⁡(40°)≈0.839\tan(40°) ≈ 0.839.)

?7 cm40°

💡 Use tan⁡(40°)=opposite7\tan(40°) = \frac{\text{opposite}}{7}. So opposite =7×tan⁡(40°)=7×0.839≈5.87= 7 \times \tan(40°) = 7 \times 0.839 ≈ 5.87 cm.

Question 4

In a right-angled triangle, the opposite side is 6 m and the hypotenuse is 10 m. What is the angle? (Use sin⁡−1(0.6)≈36.87°\sin^{-1}(0.6) ≈ 36.87°.)

6 m10 m?

💡 Use sin⁡(θ)=610=0.6\sin(\theta) = \frac{6}{10} = 0.6. So θ=sin⁡−1(0.6)≈36.87°\theta = \sin^{-1}(0.6) ≈ 36.87°.

Question 5

SOHCAHTOA reminds us which trigonometric ratios? Choose the correct interpretation of 'CAH'.

💡 'CAH' stands for Cosine = Adjacent / Hypotenuse. Always remember which side is opposite, adjacent and hypotenuse.

Common mistakes

Watch for these when working through the lesson.

  • Confusing which side is opposite and which is adjacent for a given angle. Always clearly label the three sides relative to the angle you are using.
  • Using the wrong trigonometric ratio. Make sure you know which two sides or angles you have, and which trigonometric function involves those.
  • Forgetting to use inverse trigonometry (e.g., sin⁡−1\sin^{-1}) when finding angles. You use inverse trig when the angle is unknown.
  • Rounding too early. Carry through the calculation using the full value (from your calculator) before rounding the final answer.

Related topics

These ideas fit closely with this lesson.

  • Sine rule and cosine rule in general triangles
  • Bearings and navigation
  • 3D trigonometry

Practice next

Independent practice will plug in here

This lesson builds the understanding first. Deeper adaptive practice can sit here later.