medium difficulty6 min

Using tan ratio for heights

Using the tan ratio for heights means using tan θ = opposite side / adjacent side in a right triangle to relate height and horizontal distance.

In one line: Tan links the top and the ground, so it is the height-distance ratio.

Practice This Concept

Understand it

In many class 10 questions, we know the horizontal distance and the angle of elevation or depression.
Then the tangent ratio is most useful because it connects the vertical height with the horizontal distance directly.
If the angle is θ, tan θ equals the height or opposite side divided by the horizontal or adjacent side.
This is the most common ratio in tower, tree, and building problems.

Example

If a tower is seen from 30 m away at an angle of elevation of 45 degrees, then tan 45 = height/30, so the height becomes 30 m.

Formula and rule

tan θ = opposite side / adjacent side, where opposite is the height and adjacent is the horizontal distance in standard ground-based diagrams.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1A lamp post is seen from 24 m away at an angle of elevation of 60 degrees.
2Using tan 60 = height/24, we get height = 24 tan 60 = 24 root 3 m.

Method you can reuse

1Draw the right triangle.
2Identify angle θ.
3Choose tan θ = opposite/adjacent.
4Substitute the known values.
5Solve for the unknown height or distance.
6Write the unit.

Sketch it in your answer

Draw a right triangle with the height vertical, the ground distance horizontal, and the slant line as the line of sight. Mark θ at the ground point or observer point.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is tan usually chosen in height-and-distance questions with horizontal ground?

Think first, then show answer

Tan is chosen because it directly connects the vertical height and the horizontal distance in a right triangle. That makes it the most convenient ratio for tower, building, and shadow problems.

Common wrong answer

A common wrong answer is to use sin θ = height/base without checking that the base is the adjacent side in the given triangle.

Don't mix this up

Students sometimes use sin or cos when only the height and horizontal distance are involved, or they swap opposite and adjacent sides.

Exam tip

If the question gives height and base distance, tan is usually the first ratio to test.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find the height of a tower, tree, or pole when angle and horizontal distance are given, or find the horizontal distance when angle and height are given.

1
Enough for 1 mark
Using the tan ratio for heights means using tan θ = opposite side / adjacent side in a right triangle to relate height and horizontal distance.
2
Add for 2 marks
Example: If a tower is seen from 30 m away at an angle of elevation of 45 degrees, then tan 45 = height/30, so the height becomes 30 m.
3
Add for 3 marks
Then the tangent ratio is most useful because it connects the vertical height with the horizontal distance directly.

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