Finding unknown angles
Use inverse trigonometric ratios (sin⁻¹, cos⁻¹, tan⁻¹) to find unknown angles in right-angled triangles.
Worked examples
Find an angle using sin⁻¹
Straightforward
Problem
In a right-angled triangle, the side opposite angle is cm and the hypotenuse is cm. Find to the nearest tenth of a degree.
1
Identify the known sides and the correct trig ratio.
The opposite side is cm and the hypotenuse is cm, so use .
2
Apply the inverse sine function to find .
3
Evaluate using a calculator and round to 1 decimal place.
Answer
Find the angle a ladder makes with the ground
Moderate
Problem
A m ladder leans against a vertical wall with its base m from the wall. Find the angle the ladder makes with the ground, to the nearest tenth of a degree.
1
Draw and label the right-angled triangle.
The ladder ( m) is the hypotenuse. The distance from the wall ( m) is the side adjacent to the angle with the ground.
2
Choose the correct trig ratio.
Adjacent and hypotenuse are known, so use .
3
Apply and evaluate.
Answer
The ladder makes an angle of approximately with the ground.
Find the base angle of an isosceles triangle
Challenging
Problem
An isosceles triangle has two equal sides of cm and a base of cm. Find the base angle to the nearest tenth of a degree.
1
Draw the perpendicular height from the apex to the base.
The height bisects the base, creating two right-angled triangles. Each has a hypotenuse of cm and a base of cm.
2
Identify the sides relative to the base angle .
For the base angle: adjacent cm, hypotenuse cm. Use .
3
Apply and evaluate.
Answer
The base angle is approximately .
Practise
Q1·Straightforward
In a right-angled triangle, the side opposite angle is cm and the hypotenuse is cm. Find to the nearest tenth of a degree.
Explanation
Q2·Straightforward
In a right-angled triangle, the side adjacent to angle is cm and the hypotenuse is cm. Find to the nearest tenth of a degree.
Explanation
Q3·Straightforward
In a right-angled triangle, the side opposite angle is cm and the side adjacent to is cm. Find to the nearest tenth of a degree.
Explanation
Q4·Straightforward
In a right-angled triangle, the side opposite angle is cm and the hypotenuse is cm. Find to the nearest tenth of a degree.
Explanation
Q5·Moderate
A right-angled triangle has legs of cm and cm. Find the angle that is opposite the cm side. Give your answer to the nearest tenth of a degree.
Explanation
Q6·Moderate
In a right-angled triangle, the hypotenuse is cm and the side adjacent to angle is cm. Find to the nearest tenth of a degree.
Explanation
Q7·Moderate
A m ladder leans against a vertical wall with its base m from the wall. Find the angle the ladder makes with the ground. Give your answer to the nearest tenth of a degree.
Explanation
Q8·Moderate
In a right-angled triangle, the two shorter sides are cm and cm. Find the angle that is adjacent to the cm side. Give your answer to the nearest tenth of a degree.
Explanation
Q9·Challenging
A ramp rises m vertically over a horizontal distance of m. Find the angle of inclination of the ramp to the nearest tenth of a degree.
Explanation
The rise is the opposite side and the run is the adjacent side.
Q10·Challenging
A rectangle is cm long and cm wide. Find the angle that the diagonal makes with the longer side. Give your answer to the nearest tenth of a degree.
Explanation
The diagonal, long side and short side form a right-angled triangle. The angle at the corner of the long side has:
Q11·Challenging
A road rises m for every m of horizontal distance. Find the angle of inclination of the road to the nearest tenth of a degree.
Explanation
The rise is the opposite side and the run is the adjacent side.
Q12·Challenging
An isosceles triangle has two equal sides of cm and a base of cm. Find the base angle (the angle at each end of the base) to the nearest tenth of a degree.
Explanation
Draw the perpendicular height from the apex. It bisects the base into two segments of cm each. In the right-angled triangle formed: