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Class 10 Some Applications of Trigonometry: cheat sheet
CBSE Mathematics. Find heights and distances you cannot measure directly, using angles of elevation and depression of 30°, 45° and 60°.
Key ideas
- Draw the figure first: the object is vertical, the ground horizontal, so there is a right angle at the foot.
- The angle of elevation is measured up from the horizontal; the angle of depression is measured down from the horizontal.
- The angle of elevation of A from B equals the angle of depression of B from A (alternate angles).
- tan θ links the height to the distance along the ground, which is what most problems need.
- With two angles from two points, write the height twice (once for each angle) and solve for the unknown distance.
- Keep answers exact (like 10√3 m) unless the question asks for a decimal.
Remember
| Height from distance | height = distance × tan θ |
| Distance from height | distance = height ÷ tan θ |
| Values | tan 30° = 1/√3, tan 45° = 1, tan 60° = √3 |
| Values | sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2 |
Watch out
- Measuring the angle of depression from the vertical.
- Using sin when the side you know is along the ground (use tan).
- Forgetting to add the observer's height or the height of a building the observer stands on.
Worked example
From the top of a tower, a man sees a car moving towards the foot of the tower at a steady speed. The angle of depression of the car changes from 30° to 60° in 16 minutes. How much longer will the car take to reach the tower?
Working: with height h, the distances are h√3 and h/√3; the car covers 2h/√3 in 16 minutes, and h/√3 is half of that
Answer: 8 minutes
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