# Trigonometry in the right triangle · Year 11

> Trigonometry in the right triangle, Year 11: Finding a side and more. Explanation, worked examples and free practice for the national curriculum in England.

In Trigonometry in the right triangle (Year 11) you practise 5 skills: Sine, cosine and tangent · Finding a side · Finding an angle · Special angles · The area of a triangle with sine.

In a right triangle, each acute angle has a side opposite it, a side next to it and the hypotenuse. The sine of the angle is the opposite side divided by the hypotenuse, the cosine is the adjacent side divided by the hypotenuse, and the tangent is the opposite side divided by the adjacent side. The ratios depend only on the angle, not on the size of the triangle, because all right triangles with the same acute angle are similar.

When you know a side and an angle, choose the ratio that links them with the side you want, and solve. When you know two sides, work out the ratio and find the angle with a calculator. Each question says how far to round. The sine of an angle equals the cosine of the angle that makes it up to ninety degrees, and the squares of the sine and the cosine of the same angle add up to one.

Three angles have exact values: thirty, forty-five and sixty degrees. The tangent of the angle a line makes with the x axis is its gradient. The area of a triangle is half the product of two sides times the sine of the angle between them.

**Tip:** Mark the angle, then label each side as opposite, adjacent or hypotenuse. Choose the ratio after that.

## What you learn

### Sine, cosine and tangent

- Example: (A right triangle with the right angle marked.) In triangle ABC , the angle at C is a right angle. In centimetres, AC = 24, BC = 18, AB = 30 . Find sin B as a fraction in its simplest form.
- Answer: 4/5
- Solution: Opposite the angle at B is AC , next to it is BC , and the hypotenuse is AB . So sin B = AC ÷ AB = 24/30 = 4/5 .

### Finding a side

- Example: (A right triangle with the right angle marked.) In triangle ABC , the angle at C is a right angle. In centimetres, ∢B = 25°, AB = 20 . How long is side AC ? Give your answer to 2 decimal places.
- Answer: 8.45 cm
- Solution: sin 25° = AC ÷ 20 , so AC = 20 × sin 25° ≈ 8.45 . It is about 8.45 cm long.

### Finding an angle

- Example: (A right triangle with the right angle marked.) In triangle ABC , the angle at C is a right angle. In centimetres, AC = 3, BC = 15 . Find ∢B to the nearest degree.
- Answer: 11 °
- Solution: For the angle at B we know both shorter sides, so tan B = AC ÷ BC = 3/15 . With a calculator, ∢B ≈ 11° .

### Special angles

- Example: What is the exact value of sin 60° ?
- Options: √3/3 · √3 · 1 · √3/2
- Answer: √3/2
- Solution: The exact values are sin 30° = cos 60° = 1/2 , sin 45° = cos 45° = √2/2 , sin 60° = cos 30° = √3/2 , tan 30° = √3/3 , tan 45° = 1 and tan 60° = √3 . So sin 60° = √3/2 .

### The area of a triangle with sine

- Example: (A triangle, with two of its sides and the angle between them.) In triangle ABC , in centimetres, AB = 20, AC = 5, ∢A = 30° . What is the area of the triangle?
- Answer: 25 cm²
- Solution: S = 20 × 5 × sin 30° ÷ 2 = 100 ÷ 4 = 25 . It is 25 cm².

The exercises are made by rules and the answers are worked out in code. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).

[Practise on the site](https://kixplo.com/uk/en/) · [All Year 11 topics](https://kixplo.com/uk/en/maths/year-11/)