8 min readPathway sheet2 or 3-mark question; often the first part of a case-based question

Heights and distances with one right triangle

Draw the figure, mark the right angle, then pick the ratio that links the known side to the unknown. Usually that is tan θ, which is height/distance.

Height from distanceh = d × tan θ

Slant length (string, ladder, rope)Use sin θ = height/length or cos θ = distance/length

Values to knowtan 30° = 1/√3, tan 45° = 1, tan 60° = √3; √3 = 1.732

Which ratio to use

You know	You want	Use
Distance along the ground	Height	tan θ = height/distance
Height	Distance along the ground	tan θ = height/distance
Length of a string, ladder or rope	Height	sin θ = height/length
Height	Length of the string or ladder	sin θ = height/length
Length of a ladder	Distance of its foot from the wall	cos θ = distance/length

Method for one right triangle

1
Read the question twice and draw the figure.
2
Mark the right angle, the given angle, the known side, and the unknown h or x.
3★
Choose the ratio that contains both the known and the unknown side.
4
Put in the value of the ratio from the table and solve.
5
Use √3 = 1.732 if asked, round to 2 decimals and write the unit.

★ key step

Height from a distance (tan)

Problem

From a point on level ground 25 m from the foot of a temple gopuram, the angle of elevation of its top is 60°. Find the height of the gopuram. (Use √3 = 1.732)

Working

Let AB be the gopuram and C the point. BC = 25 m, ∠ACB = 60°, ∠B = 90°.
We know the adjacent side and want the opposite side, so use tan.
tan 60° = AB/BC, so √3 = AB/25.
AB = 25√3 = 25 × 1.732 = 43.3 m.

Answer

43.3 m

Word problem: a ladder (sin and cos)

Problem

A ladder 6 m long leans against a wall and makes 60° with the ground. How high up the wall does it reach, and how far is its foot from the wall? (Use √3 = 1.732)

Working

Let AC = 6 m be the ladder, C its foot and AB the wall. ∠ACB = 60°, ∠B = 90°.
Height: sin 60° = AB/AC, so √3/2 = AB/6.
AB = 3√3 = 3 × 1.732 = 5.196 m ≈ 5.2 m.
Foot: cos 60° = BC/AC, so 1/2 = BC/6.
BC = 3 m.

Answer

It reaches about 5.2 m up the wall; its foot is 3 m from the wall.

Common slips

Using sin when the ground distance (not the slant length) is given. Use tan.
Forgetting to add the observer's eye height at the end.
Measuring the angle from the vertical instead of the horizontal.
Stopping at 25√3 when the question asks for a decimal value.
Leaving out the unit (m) in the final answer.

In real life

Before Diwali, you can find the length of a light string to the terrace from just the angle and the house height. Electricians and painters use the same ladder ratios.

Exam angle

1 marks: “A ladder makes 60° with the ground and its foot is 2 m from the wall. Find the length of the ladder.” cos 60° = 2/L, so 1/2 = 2/L and L = 4 m.

2 marks: “A kite is flying on a 100 m long string that makes 30° with the ground. Assuming the string is straight, find the height of the kite.” sin 30° = h/100, so h = 100 × 1/2 = 50 m.

3 marks: “The angle of elevation of the top of a 30 m tall water tank from a point on the ground is 30°. How far is the point from the foot of the tank? (Use √3 = 1.732)” tan 30° = 30/d, so d = 30√3 = 30 × 1.732 = 51.96 m.

Remember it

Ground known? Use tan. String or ladder known? Use sin or cos.