Trigonometry in the right triangle · Grade 10 · 4 units
In Trigonometry in the right triangle (Grade 10 · 4 units) you practise 9 skills: Sine, cosine and tangent · Finding a side · Finding an angle · Special angles · Links between the ratios · Gradient and angle · The area of a triangle with sine · Triangles inside shapes · The values of the ratios. 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
For example: 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 and solution
The answer: 4/5
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
For example: 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 and solution
The answer: 8.45 cm
sin 25° = AC ÷ 20 , so AC = 20 × sin 25° ≈ 8.45 . It is about 8.45 cm long.
Finding an angle
For example: In triangle ABC , the angle at C is a right angle. In centimetres, AC = 3, BC = 15 . Find ∢B to the nearest degree.
Answer and solution
The answer: 11 °
For the angle at B we know both shorter sides, so tan B = AC ÷ BC = 3/15 . With a calculator, ∢B ≈ 11° .
Special angles
For example: What is the exact value of sin 60° ?
- √3/3
- √3
- 1
- √3/2
Answer and solution
The answer: √3/2
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 .
Links between the ratios
For example: The angle α is acute, and sin α = 0.6 . Find tan α as a fraction in its simplest form.
Answer and solution
The answer: 3/4
cos α = √(1 − 0.6²) = √(1 − 0.36) = √0.64 = 0.8 . So tan α = 0.6 ÷ 0.8 = 3/4 .
Gradient and angle
For example: A rising line makes an angle of 45° with the positive direction of the x axis. What is its gradient?
- √3
- √3/3
- √3/2
- 1
Answer and solution
The answer: 1
The gradient is tan 45° = 1 .
The area of a triangle with sine
For example: In triangle ABC , in centimetres, AB = 20, AC = 5, ∢A = 30° . What is the area of the triangle?
Answer and solution
The answer: 25 cm²
S = 20 × 5 × sin 30° ÷ 2 = 100 ÷ 4 = 25 . It is 25 cm².
Triangles inside shapes
For example: In triangle ABC , AB = AC = 6 cm, and ∢B = 75° . The line AD is the height to the base BC . How long is AD ? Give your answer to 2 decimal places.
Answer and solution
The answer: 5.8 cm
In the right triangle ABD , AD = 6 × sin 75° ≈ 5.8 . It is about 5.8 cm long.
The values of the ratios
For example: As the acute angle α grows, what happens to tan α ?
- It gets smaller
- It stays the same
- It gets larger
Answer and solution
The answer: It gets larger
As the acute angle grows with the hypotenuse kept the same, the side opposite it grows and the side next to it shrinks. So the sine and the tangent grow, and the cosine shrinks. tan 20° ≈ 0.364 , tan 70° ≈ 2.747 .
Not on the site yet
Parts of this topic that are not built yet: Proving the special values · The sine rule and the cosine rule · Radians.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).