Finding unknown sides
Use SOHCAHTOA to find an unknown side in a right-angled triangle given one angle and one side.
Worked examples
Finding the opposite side given the hypotenuse
Straightforward
Problem
A right-angled triangle has a hypotenuse of cm and an angle of . Find the length of the side opposite the angle, correct to 2 decimal places.
1
Identify the known information and the unknown side.
2
Choose the ratio that links the opposite side and the hypotenuse. SOH: .
3
Rearrange to make the subject.
4
Evaluate using a calculator and round to 2 decimal places.
Answer
cm
Finding the hypotenuse given the adjacent side
Moderate
Problem
In a right-angled triangle, the side adjacent to a angle is m. Find the hypotenuse, correct to 2 decimal places.
1
Identify the known information and the unknown.
2
Choose the ratio that links the adjacent side and the hypotenuse. CAH: .
3
Rearrange to make the subject. Multiply both sides by , then divide both sides by .
4
Evaluate using a calculator and round to 2 decimal places.
Answer
m
Finding a side in a real-world context
Challenging
Problem
A cable car wire runs from the base of a mountain to the summit. The wire makes an angle of with the horizontal and the horizontal distance from base to summit is m. Find the length of the cable wire, correct to 2 decimal places.
1
Draw and label a right-angled triangle. The horizontal distance ( m) is the adjacent side, the wire is the hypotenuse, and the angle with the horizontal is .
2
Choose the ratio that links the adjacent side and the hypotenuse. CAH: .
3
Rearrange to make the subject.
4
Evaluate using a calculator and round to 2 decimal places.
Answer
m
Practise
Q1·Straightforward
In a right-angled triangle, the hypotenuse is cm and one angle is . Find the length of the side opposite the angle. Give your answer correct to 2 decimal places.
Explanation
Using SOH: , so cm.
Q2·Straightforward
In a right-angled triangle, the hypotenuse is cm and one angle is . Find the length of the side adjacent to the angle. Give your answer correct to 2 decimal places.
Explanation
Using CAH: , so cm.
Q3·Straightforward
In a right-angled triangle, the hypotenuse is m and one angle is . Find the length of the side opposite the angle. Give your answer correct to 2 decimal places.
Explanation
Using SOH: , so m.
Q4·Straightforward
In a right-angled triangle, the hypotenuse is m and one angle is . Find the length of the side adjacent to the angle. Give your answer correct to 2 decimal places.
Explanation
Using CAH: , so m.
Q5·Moderate
In a right-angled triangle, the side opposite a angle is cm. Find the length of the hypotenuse. Give your answer correct to 2 decimal places.
Explanation
Using SOH: , so cm.
Q6·Moderate
In a right-angled triangle, the side adjacent to a angle is m. Find the length of the side opposite the angle. Give your answer correct to 2 decimal places.
Explanation
Using TOA: , so m.
Q7·Moderate
In a right-angled triangle, the side adjacent to a angle is cm. Find the length of the hypotenuse. Give your answer correct to 2 decimal places.
Explanation
Using CAH: , so cm.
Q8·Moderate
In a right-angled triangle, the side opposite a angle is m. Find the length of the adjacent side. Give your answer correct to 2 decimal places.
Explanation
Using TOA: , so m.
Q9·Challenging
A flagpole is m tall. A support wire runs from the top of the flagpole to a point on the ground, making an angle of with the horizontal ground. Find the length of the support wire. Give your answer correct to 2 decimal places.
Explanation
The flagpole (opposite) is m and the wire (hypotenuse) makes a angle with the ground. Using SOH: , so m.
Q10·Challenging
A ramp is m long and makes an angle of with the horizontal ground. Find the vertical height gained by walking up the full length of the ramp. Give your answer correct to 2 decimal places.
Explanation
The ramp (hypotenuse) is m and the angle with the ground is . The vertical height is the opposite side. Using SOH: , so m.
Q11·Challenging
A right-angled triangle has an angle of . The side adjacent to this angle is cm. Find the length of the side opposite the angle. Give your answer correct to 2 decimal places.
Explanation
Using TOA: , so cm.
Q12·Challenging
A boat is m from the base of a cliff. The angle from the boat up to the top of the cliff is . Find the height of the cliff. Give your answer correct to 2 decimal places.
Explanation
The horizontal distance (adjacent) is m and the angle is . The cliff height is the opposite side. Using TOA: , so m.