The shadow of a vertical tower on level ground is equal to its height. The angle of elevation of the Sun is
- (a)\(30^\circ\)
- (b)\(45^\circ\)
- (c)\(60^\circ\)
- (d)\(90^\circ\)
Some Applications of Trigonometry Class 10: notes and important questions
Revision notes, 37 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.
- 37 questions
- 10 multiple choice, 3 assertion–reason, 8 very short answer, 8 short answer, 5 long answer, 3 case study
- About 10 hours to master
Trigonometry unit: 12 of 80 theory marks (Introduction to Trigonometry and Some Applications of Trigonometry).
Revision notes
Key terms
- Line of sight: the line from the observer's eye to the object viewed.
- Angle of elevation: the angle between the horizontal line and the line of sight when the object is above the horizontal through the eye (you raise your head).
- Angle of depression: the angle between the horizontal line and the line of sight when the object is below the horizontal (you lower your head).
- The angle of elevation of \(Q\) from \(P\) equals the angle of depression of \(P\) from \(Q\). They are alternate angles between parallel horizontal lines.
Standard method
- Draw a neat figure. Mark vertical objects as vertical lines, the ground as a horizontal line, and put a right angle at the foot.
- Mark the given angles from the horizontal. For an angle of depression, draw the horizontal through the observer first.
- Choose the ratio that links the known side with the unknown one. Usually \(\tan\theta=\dfrac{\text{height}}{\text{horizontal distance}}\).
- With two triangles, write one equation from each, then eliminate the common unknown (usually the horizontal distance).
- CBSE is a no-calculator exam: leave answers in surd form (rationalise the denominator). Use \(\sqrt3=1.732\) or \(\sqrt2=1.414\) only when the question gives the value, as board papers do (“Take \(\sqrt3=1.73\)”).
Useful values: \(\tan30^\circ=\frac{1}{\sqrt3},\ \tan45^\circ=1,\ \tan60^\circ=\sqrt3\); \(\sin30^\circ=\frac12,\ \sin45^\circ=\frac{1}{\sqrt2},\ \sin60^\circ=\frac{\sqrt3}{2}\).
Worked example 1 (one triangle)
A 12 m ladder leans against a wall and makes \(60^\circ\) with the ground. How high up the wall does it reach?
\(\sin60^\circ=\dfrac{h}{12}\Rightarrow h=12\cdot\frac{\sqrt3}{2}=6\sqrt3\) m.
Worked example 2 (observer moves)
The angle of elevation of the top of a tower is \(30^\circ\). After walking 40 m towards the tower it becomes \(60^\circ\). Find the height.
Let the height be \(h\) and the final distance \(x\). Then \(h=x\sqrt3\) and \(h=\dfrac{x+40}{\sqrt3}\). So \(3x=x+40\), giving \(x=20\) and \(h=20\sqrt3\) m.
Worked example 3 (elevation and depression from one point)
From the top of an 8 m building, the angle of elevation of the top of a pole is \(45^\circ\) and the angle of depression of its foot is \(30^\circ\). Find the height of the pole.
Horizontal distance \(d=8\cot30^\circ=8\sqrt3\). The part of the pole above the building's level is \(d\tan45^\circ=8\sqrt3\). Pole \(=8+8\sqrt3=8(1+\sqrt3)\) m.
Common errors
- Measuring the angle of depression from the vertical instead of the horizontal.
- Forgetting the observer's eye height when a person's height is given.
- Using \(\sin\) when the hypotenuse is not involved. Pick the ratio from the two sides you actually have.
- Rounding \(\sqrt3\) too early. Keep surds until the last line.
Board-exam tips
- A correct, labelled diagram usually carries a mark in 3- and 5-mark questions. Always draw one.
- Syllabus limits: at most two right triangles, and angles of elevation/depression of only 30°, 45° and 60°.
- State units and give the answer in the form asked (surd or decimal).
Topics in this chapter: Angle of elevation · Angle of depression · Two-triangle problems · Angle of elevation and depression.
Route to 95: four steps
Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.
Step 1
Secure the basics
You can draw the right triangle for a height-and-distance problem and use tan, sin or cos with 30°, 45° and 60°.
Step 2
Board standard
You can solve angle-of-depression problems and the standard two-triangle problems where the observer moves.
Step 3
Full marks on long answers
You can write full-marks 5-mark answers with a labelled figure, two equations and a final answer with units.
Step 4
95+ stretch (HOTS)
You can handle mixed elevation-and-depression problems, moving objects and speed, and the unusual two-triangle set-ups.
Practice questions
Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 29 of the 37 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.
A straight zip-line cable, 80 m long, runs from the top of a platform down to a point on level ground and makes an angle of \(30^\circ\) with the ground. The height of the platform is
- (a)\(40\) m
- (b)\(40\sqrt3\) m
- (c)\(80\sqrt3\) m
- (d)\(\frac{40}{\sqrt3}\) m
A 14 m ladder leans against a wall and makes an angle of \(60^\circ\) with the ground. The distance of its foot from the wall is
- (a)\(7\sqrt3\) m
- (b)\(7\) m
- (c)\(14\sqrt3\) m
- (d)\(\frac{7}{\sqrt3}\) m
Where marks are lost in Some Applications of Trigonometry
- No figure, or an unlabelled one. Fix: draw it every time, with heights, distances and angles marked; it usually carries B1 in 3- and 5-mark questions.
- Measuring the angle of depression from the vertical. Fix: it is measured from the horizontal at the observer's eye; it equals the angle of elevation from the other end (alternate angles).
- Ignoring the observer's height. Fix: subtract the eye height from the building or tower before using tan, and add it back if the question asks for the full height.
- Rounding √3 too early and losing the A1. Fix: keep surds until the final line, then use the given value (1.732).
- Using sin when the hypotenuse is not involved. Fix: pick the ratio from the two sides you have: opposite and adjacent means tan.
- Answers without units or not in the form asked. Fix: finish with a sentence, e.g. 'Height of the tower = 20√3 m' (or '= 34.64 m' only when the question gives √3 = 1.732).
Some Applications of Trigonometry in our sample papers
Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).
- Sample paper 1 (Board standard): questions 11, 34
- Sample paper 2 (Board standard): questions 13, 36
- Sample paper 3 (Above board): question 34
- Sample paper 4 (Above board): questions 13, 34
- Sample paper 5 (95+ challenge): question 36