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Shadow-based reasoning

Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.

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Student-friendly explanation

When sunlight falls on a pole, tree, or building, the object, its shadow, and the sun ray form a right triangle. The shadow becomes the horizontal side, and the height becomes the vertical side. This makes tan the most useful ratio in many shadow questions. These problems are very common because they connect trigonometry with everyday observation.

How to write this in exams

  1. 1

    Start with the exact idea

    Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.

  2. 2

    Then show how to use it

    1. Draw the object and its shadow. 2. Mark the sun angle. 3. Identify height and shadow as opposite and adjacent. 4. Use tan theta. 5. Solve. 6. Write the answer with units.

  3. 3

    Add one concrete example

    A tree casts a 12 m shadow when the angle of elevation of the sun is 30 degrees. Then tan 30 = height/12, so the tree height can be found.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to use the shadow as the hypotenuse. The hypotenuse is the sun ray, not the ground shadow.

Definition

Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.

Example

A tree casts a 12 m shadow when the angle of elevation of the sun is 30 degrees. Then tan 30 = height/12, so the tree height can be found.

Rule to remember

On level ground, tan theta = height / shadow length when the light ray and ground form a right triangle.

Memory hook

Shadow sits on the ground, so it is the adjacent side.

Examples and method

Worked example

A flagpole casts a 9 m shadow and the sun's angle of elevation is 45 degrees. Since tan 45 = height/9, the height is 9 m.

Method to apply

1. Draw the object and its shadow. 2. Mark the sun angle. 3. Identify height and shadow as opposite and adjacent. 4. Use tan theta. 5. Solve. 6. Write the answer with units.

Diagram support

Draw the object vertical, the shadow horizontal on the ground, and the sun ray as the slant side.

How CBSE asks it

Find height from shadow length and sun angle, or find shadow length from height and sun angle.

Avoid common mistakes

Common confusion

Students sometimes use the shadow as the slant side instead of the horizontal side.

Common wrong answer

A common wrong answer is to use the shadow as the hypotenuse. The hypotenuse is the sun ray, not the ground shadow.

Exam tip

In shadow questions, always treat the shadow length as the ground side unless the question says the ground is sloping.

Quick check

In a shadow problem on level ground, which side usually represents the shadow length?

The shadow length usually represents the horizontal side on the ground. That is why tan is often used to relate the shadow and the height.

Answer writing and exam use

1-mark answer

Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance.

2-mark answer

Shadow-based reasoning uses the length of a shadow and the angle of elevation of the sun or light source to find a height or distance. On level ground, tan theta = height / shadow length when the light ray and ground form a right triangle. A tree casts a 12 m shadow when the angle of elevation of the sun is 30 degrees. Then tan 30 = height/12, so the tree height can be found.

3-mark answer

When sunlight falls on a pole, tree, or building, the object, its shadow, and the sun ray form a right triangle. The shadow becomes the horizontal side, and the height becomes the vertical side. This makes tan the most useful ratio in many shadow questions. These problems are very common because they connect trigonometry with everyday observation. On level ground, tan theta = height / shadow length when the light ray and ground form a right triangle. A flagpole casts a 9 m shadow and the sun's angle of elevation is 45 degrees. Since tan 45 = height/9, the height is 9 m. Find height from shadow length and sun angle, or find shadow length from height and sun angle. A common wrong answer is to use the shadow as the hypotenuse. The hypotenuse is the sun ray, not the ground shadow.
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