# Trigonometry in any triangle · Grade 10 · 5 units

> Trigonometry in any triangle, Grade 10 · 5 units: The sine rule and more. Explanation, worked examples and free practice following the Israeli curriculum.

In Trigonometry in any triangle (Grade 10 · 5 units) you practise 7 skills: The area of a triangle with the sine · The sine rule · The cosine rule · One triangle or two · Angles with the same sine · The largest value without a derivative · Solving a triangle in two steps.

The area of any triangle, not only a right triangle, is half the product of two sides times the sine of the angle between them. For an obtuse angle, use the fact that an angle and the angle that makes it up to 180 degrees have the same sine. So the same area can come from two different angles, an acute one and an obtuse one.

The sine rule says that in every triangle a side over the sine of the angle opposite it is the same for all three sides, and it equals the diameter of the circle through the corners. The cosine rule finds a side from the other two sides and the angle between them, or an angle from the three sides. When two sides and an angle that is not between them are given, there can be two triangles, one or none.

When the question asks for an exact answer, keep the roots: for example, the sine of 60 degrees is root three over two. When it asks for rounding, use a calculator and round only at the end. A sine is never more than one, so you can find a largest value without a derivative.

A model proof of the sine rule: write the area of the triangle in three ways, each time with two sides and the sine of the angle between them. The three expressions are equal. Divide each of them by half the product of all three sides, and what is left is the sine of each angle over the side facing it, the same number for all three angles. Turn the fractions over and you have the sine rule. The cosine rule is proved with the altitude from one corner: Pythagoras’ theorem in the two right triangles the altitude makes, and the cosine for the part of the base next to the angle.

**Tip:** Before you calculate, mark what is given on the triangle: two sides and the angle between them lead to the cosine rule, and a side with the angle opposite it leads to the sine rule.

## What you learn

### The area of a triangle with the sine

- Example: (A triangle with the given sides and angles.) In triangle ABC , in centimetres, AB = 4, AC = 12, ∢A = 30° . What is the area of the triangle?
- Answer: 12 cm²
- Solution: The sine of 30° is 1/2 , so S = 4 × 12 × 1/2 ÷ 2 = 12 cm².

### The sine rule

- Example: (A triangle with the given sides and angles.) In triangle ABC you are told that AC = 6, ∢B = 30°, ∢C = 100° . How long is the side AB in cm? Give your answer to 2 decimal places.
- Answer: 11.82 cm
- Solution: The side AB is opposite the angle at C , and the side AC is opposite the angle at B : AB / sin 100° = 6 / sin 30° , so AB = 6 × sin 100° / sin 30° ≈ 11.82 cm.

### The cosine rule

- Example: (A triangle with the given sides and angles.) In triangle ABC , in centimetres, AB = 3, AC = 8, ∢A = 60° . How long is BC ?
- Answer: 7 cm
- Solution: The cosine of 60° is 1/2 , so BC² = 3² + 8² − 2 × 3 × 8 × 1/2 = 9 + 64 − 24 = 49 . So BC = 7 cm.

### One triangle or two

- Example: In triangle ABC , in centimetres, AC = 6, BC = 3, ∢A = 70° . How many such triangles are there?
- Options: One triangle · Two triangles · There is no such triangle
- Answer: There is no such triangle
- Solution: By the sine rule, sin B = AC × sin A / BC = 6 × sin 70° / 3 ≈ 1.8794 . A sine is never more than 1, so there is no such triangle.

### Angles with the same sine

- Example: You are told that sin α = sin 20° , and the angle α is obtuse. What is α ?
- Answer: 160 °
- Solution: The angle 20° is acute, and the angle that makes it up to 180° has the same sine: α = 180° − 20° = 160° .

### The largest value without a derivative

- Example: What is the largest value of 4 sin 2α ?
- Answer: 4
- Solution: A sine is never more than 1, so 4 sin 2α ≤ 4 . It is equal when 2α = 90° , that is when α = 45° . The largest value is 4 .

### Solving a triangle in two steps

- Example: (A triangle with the given sides and angles.) In triangle ABC , in centimetres, AB = 5, AC = 12, BC = 9 . What is the area of the triangle? Give your answer to 2 decimal places.
- Answer: 20.4 cm²
- Solution: By the cosine rule, cos A = (5² + 12² − 9²) / (2 × 5 × 12) ≈ 0.7333 , so sin A = √(1 − cos²A) ≈ 0.6799 . The area is S = 5 × 12 × sin A ÷ 2 ≈ 20.4 cm².

## Not on the site yet

- Proving the area formula, the sine rule and the cosine rule
- Geometric proofs with trigonometry

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).

[Practise on the site](https://kixplo.com/il/en/) · [All Grade 10 · 5 units topics](https://kixplo.com/il/en/math/grade-10-5u/)