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Using tan ratio for heights

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

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Student-friendly explanation

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 theta, tan theta 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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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

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.

Rule to remember

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

Memory hook

Tan links the top and the ground, so it is the height-distance ratio.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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

2-mark answer

Using the tan ratio for heights means using tan theta = opposite side / adjacent side in a right triangle to relate height and horizontal distance. tan theta = opposite side / adjacent side, where opposite is the height and adjacent is the horizontal distance in standard ground-based diagrams. 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-mark answer

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 theta, tan theta 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. tan theta = opposite side / adjacent side, where opposite is the height and adjacent is the horizontal distance in standard ground-based diagrams. A lamp post is seen from 24 m away at an angle of elevation of 60 degrees. Using tan 60 = height/24, we get height = 24 tan 60 = 24 root 3 m. 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. A common wrong answer is to use sin theta = height/base without checking that the base is the adjacent side in the given triangle.
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