Right Triangle Trigonometry — Practice
One angle and one side are enough to find every other part of a right triangle.
What these questions cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Base — The number being multiplied. In 53 the base is 5.
- Hypotenuse — The side opposite the right angle — always the longest.
- Simplest radical form — No perfect square factors left under the root.
- Adjacent — The side next to that angle which is not the hypotenuse.
- Sine — sin θ = oppositehypotenuse.
- Tangent — tan θ = oppositeadjacent.
- Angle of elevation — The angle up from horizontal to a line of sight.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
In a right triangle, which ratio is oppositehypotenuse?
Worked answer
sine
sin θ = OH
Problem 2
Which ratio is oppositeadjacent?
Worked answer
tangent
tan θ = OA
Problem 3
Which side of a right triangle is always the longest?
Worked answer
hypotenuse
The hypotenuse is opposite the right angle and always longest.
Problem 4
A right triangle has hypotenuse 10 and an angle of 30°. Find the opposite side.
Worked answer
5
sin 30° = x10
x = 10 × 0.5 = 5
Problem 5
tan θ = 1. Find θ.
Worked answer
45
θ = tan-1(1) = 45°
Problem 6
A ladder 6 m long leans at 70°. How high does it reach? Round to one decimal place.
Worked answer
5.6
sin 70° = h6
h = 6 × 0.9397 ≈ 5.6 m
Problem 7
cos 40° = 8x. Find x to one decimal place.
Worked answer
10.4
x = 8cos 40° = 80.766
≈ 10.4
Problem 8
A ramp rises 1.5 m over a run of 9 m. Find the angle of elevation to one decimal place.
Worked answer
9.5
tan θ = 1.59 = 0.1667
θ ≈ 9.5°
Problem 9
From a cliff 40 m high the angle of depression to a boat is 25°. Find the distance to the base, to one decimal place.
Worked answer
85.8
tan 25° = 40d
d = 400.4663 ≈ 85.8 m
Problem 10
sin-1(0.5) means
- the angle whose sine is 0.5
- 1sin 0.5
- sin(0.5) doubled
- half of sin
Worked answer
the angle whose sine is 0.5
It undoes the sine.
sin-1(0.5) = 30°.
Problem 11
In a 45-45-90 triangle each leg is 7. Find the hypotenuse in simplest radical form.
Worked answer
7sqrt(2)
Hypotenuse = 7√2
Problem 12
In a 30-60-90 triangle the short leg is 5. Find the hypotenuse.
Worked answer
10
Hypotenuse is twice the short leg.
= 10
Problem 13
A kite string 50 m long makes 58° with the ground. How high is the kite, to the nearest metre?
Worked answer
42
sin 58° = h50
h = 50 × 0.848 ≈ 42 m
Problem 14
A right triangle has legs 5 and 12. Find the smaller acute angle to one decimal place.
Worked answer
22.6
tan θ = 512
θ ≈ 22.6°
Problem 15
From a point 30 m from a tower, the angle of elevation to the top is 52°. Find the tower height to the nearest metre.
Worked answer
38
tan 52° = h30
h = 30 × 1.2799 ≈ 38 m
Problem 16
In a 30-60-90 triangle the hypotenuse is 14. Find the longer leg in simplest radical form.
Worked answer
7sqrt(3)
Short leg = 7
Long leg = 7√3