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Some Applications of Trigonometry

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter applies trigonometric ratios to solve real-world problems involving heights and distances. It introduces the concepts of line of sight, angle of elevation, and angle of depression, which are used to measure inaccessible heights or distances using right triangle trigonometry. The one big idea a student must walk away with is that by knowing one side and one acute angle of a right triangle, you can find any other side using appropriate trigonometric ratios (sine, cosine, tangent, or cotangent).
Key Concepts (11)
Line of sight
The line drawn from the eye of an observer to the point in the object viewed by the observer.
e.g. When a student looks at the top of a minar, the straight line from the student's eye to the top of the minar is the line of sight.
Angle of elevation
The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level, i.e., the case when we raise our head to look at the object.
e.g. Looking up at the top of a chimney from a point on the ground, the angle between the horizontal and the line of sight is the angle of elevation.
Angle of depression
The angle formed by the line of sight with the horizontal when the point is below the horizontal level, i.e., the case when we lower our head to look at the point being viewed.
e.g. A girl sitting on a balcony looking down at a flower pot on the ground; the angle between the horizontal and her line of sight is the angle of depression.
Horizontal level
The imaginary line parallel to the ground at the eye level of the observer.
e.g. When standing on level ground, the horizontal line from your eye straight ahead is the horizontal level.
Observer
The person from whose eye the line of sight is drawn.
e.g. The student looking at the minar is the observer.
Point being viewed
The specific point on the object that the observer is looking at.
e.g. The top of the minar or the bottom of the building being observed.
Height of the object
The vertical distance from the base of the object to its top, often found using trigonometric ratios.
e.g. The height of a tower is found by using tan of the angle of elevation and the distance from the foot.
Distance from the object
The horizontal distance between the observer and the foot of the object.
e.g. The distance of a point on the ground from the foot of a tower is 15 m.
Right triangle
A triangle with one angle equal to 90°, used to model height and distance problems.
e.g. Triangle ABC with right angle at B, where AB is the tower and BC is the distance from the foot.
Trigonometric ratio
A ratio of two sides of a right triangle, such as sine, cosine, tangent, or cotangent, used to relate angles and sides.
e.g. tan 60° = AB/BC is used to find the height of a tower when the distance and angle are known.
Sun's altitude
The angle of elevation of the sun above the horizon, which determines the length of shadows.
e.g. When the sun's altitude is 60°, the shadow of a tower is shorter than when it is 30°.
Formulas & Equations (13)
tan A = BC/AB
Reads as: tan of angle A equals opposite side BC divided by adjacent side AB
Variables: A = angle of elevation or depression; BC = opposite side (height); AB = adjacent side (distance)
Use when: When you need to find the height or distance in a right triangle and you know the angle and one side.
⚠ Students often confuse which side is opposite and which is adjacent, especially when the triangle is not drawn to scale.
cot A = AB/BC
Reads as: cot of angle A equals adjacent side AB divided by opposite side BC
Variables: A = angle; AB = adjacent side (distance); BC = opposite side (height)
Use when: When you need to find the distance or height using the cotangent ratio, often as an alternative to tan.
⚠ Students forget that cot A = 1/tan A and may incorrectly invert the sides.
sin 60° = BD/BC
Reads as: sine of 60 degrees equals opposite side BD divided by hypotenuse BC
Variables: BD = height to be reached (3.7 m); BC = length of ladder (hypotenuse)
Use when: When the problem involves the hypotenuse and the opposite side, such as finding the length of a ladder or a rope.
⚠ Students may use tan instead of sin when the hypotenuse is involved, leading to wrong calculations.
cot 60° = DC/BD
Reads as: cot of 60 degrees equals adjacent side DC divided by opposite side BD
Variables: DC = distance from foot of pole to ladder; BD = height to be reached
Use when: When finding the horizontal distance from the foot of a pole to the ladder using the angle of inclination.
⚠ Students may mistakenly use tan 60° = BD/DC, which gives the same numerical value but requires careful substitution.
tan 45° = AE/DE
Reads as: tan of 45 degrees equals opposite side AE divided by adjacent side DE
Variables: AE = height of chimney above eye level; DE = horizontal distance from observer to chimney
Use when: When the angle of elevation is 45°, making tan 45° = 1, so height equals distance.
⚠ Students forget to add the observer's height to AE to get the total height of the object.
tan 30° = AB/AP
Reads as: tan of 30 degrees equals opposite side AB divided by adjacent side AP
Variables: AB = height of building (10 m); AP = distance from point P to building
Use when: When finding the distance from a point to a building given the angle of elevation and building height.
⚠ Students may incorrectly rationalize the denominator or forget that tan 30° = 1/√3.
tan 45° = AD/AP
Reads as: tan of 45 degrees equals opposite side AD divided by adjacent side AP
Variables: AD = height of building plus flagstaff; AP = distance from point P to building
Use when: When finding the height of a flagstaff on top of a building using two angles of elevation.
⚠ Students may forget that AD = AB + DB and not add the building height correctly.
tan 60° = h/x
Reads as: tan of 60 degrees equals height h divided by shadow length x
Variables: h = height of tower; x = length of shadow when sun's altitude is 60°
Use when: When comparing shadow lengths at different sun altitudes to find the height of a tower.
⚠ Students may mix up which shadow length corresponds to which angle.
tan 30° = h/(40 + x)
Reads as: tan of 30 degrees equals height h divided by shadow length (40 + x)
Variables: h = height of tower; (40 + x) = length of shadow when sun's altitude is 30°
Use when: When the shadow is 40 m longer at 30° than at 60°, forming two equations to solve for h and x.
⚠ Students may forget to add the 40 m to x, using x instead of (40 + x).
tan 30° = PD/BD
Reads as: tan of 30 degrees equals opposite side PD divided by adjacent side BD
Variables: PD = height difference between buildings; BD = distance between buildings
Use when: When finding the height of a multi-storeyed building using angles of depression to the top and bottom of another building.
⚠ Students may confuse angle of depression with angle of elevation and set up the triangle incorrectly.
tan 45° = PC/AC
Reads as: tan of 45 degrees equals opposite side PC divided by adjacent side AC
Variables: PC = height of multi-storeyed building; AC = distance between buildings
Use when: When the angle of depression to the bottom of a building is 45°, making height equal to distance.
⚠ Students may forget that PC = PD + DC and DC = AB = 8 m.
tan 30° = PD/AD
Reads as: tan of 30 degrees equals opposite side PD divided by adjacent side AD
Variables: PD = height of bridge above bank (3 m); AD = distance from bridge to one bank
Use when: When finding the width of a river from a bridge using angles of depression to opposite banks.
⚠ Students may forget that the angle of depression from the bridge equals the angle of elevation from the bank.
tan 45° = PD/BD
Reads as: tan of 45 degrees equals opposite side PD divided by adjacent side BD
Variables: PD = height of bridge (3 m); BD = distance from bridge to the other bank
Use when: When the angle of depression to one bank is 45°, making the distance equal to the height.
⚠ Students may add AD and BD incorrectly or forget to convert units.
How It Works
Finding the height of an object using angle of elevation
  1. Step 1: Draw a right triangle with the object as the vertical side, the horizontal distance as the base, and the line of sight as the hypotenuse.
  2. Step 2: Identify the known values: angle of elevation and horizontal distance (or height).
  3. Step 3: Choose the appropriate trigonometric ratio (tan or cot) that relates the known and unknown sides.
  4. Step 4: Substitute the known values and solve for the unknown side.
  5. Step 5: If the observer's height is given, add it to the calculated height to get the total height of the object.
💡 Always check whether the observer's height is included in the problem. If the observer is standing on the ground, their eye level is not at ground level unless stated otherwise.
Finding the distance to an object using angle of elevation
  1. Step 1: Draw a right triangle with the object's height as the vertical side and the distance as the base.
  2. Step 2: Identify the known values: angle of elevation and height of the object.
  3. Step 3: Choose tan or cot ratio that relates the known height to the unknown distance.
  4. Step 4: Substitute and solve for the distance.
  5. Step 5: If the observer's height is given, subtract it from the total height before using the ratio.
💡 When the angle is 45°, tan 45° = 1, so height equals distance — a common shortcut but only valid for 45°.
Solving problems with two angles of elevation (e.g., flagstaff on a building)
  1. Step 1: Draw two right triangles sharing the same base: one for the building alone, one for the building plus flagstaff.
  2. Step 2: Use the first triangle (building only) to find the distance from the point to the building.
  3. Step 3: Use the second triangle (building + flagstaff) with the same distance to find the total height.
  4. Step 4: Subtract the building height from the total height to get the flagstaff height.
💡 The distance from the point to the building is the same in both triangles — do not recalculate it separately.
Solving shadow problems with two sun altitudes
  1. Step 1: Draw two right triangles with the same tower height but different shadow lengths.
  2. Step 2: Let the shorter shadow be x and the longer shadow be x + given difference.
  3. Step 3: Write two equations using tan of each angle: tan 60° = h/x and tan 30° = h/(x + 40).
  4. Step 4: Solve the system of equations to find h and x.
  5. Step 5: The height of the tower is h.
💡 The shadow is longer when the sun's altitude is lower (smaller angle). Do not reverse the angles.
Solving problems with angles of depression
  1. Step 1: Draw the diagram with the observer at the top of a higher object looking down at a lower object.
  2. Step 2: Use the property that the angle of depression from the observer equals the angle of elevation from the lower point (alternate interior angles).
  3. Step 3: Form right triangles using the horizontal line from the observer's eye and the vertical line from the lower object.
  4. Step 4: Apply trigonometric ratios to find unknown heights or distances.
  5. Step 5: If two angles of depression are given (to top and bottom of a building), set up two equations and solve.
💡 The angle of depression is always measured from the horizontal downward. Do not confuse it with the angle inside the triangle at the observer's eye.
Finding the width of a river using angles of depression from a bridge
  1. Step 1: Draw the bridge at height h above the river, with points A and B on opposite banks.
  2. Step 2: The angles of depression to A and B are given (e.g., 30° and 45°).
  3. Step 3: In right triangle APD (with point A), use tan 30° = h/AD to find AD.
  4. Step 4: In right triangle PBD (with point B), use tan 45° = h/BD to find BD.
  5. Step 5: The width of the river AB = AD + BD.
💡 The two triangles share the same height h but have different base lengths. Do not assume the distances are equal.
Exam Traps — Don't Make These Mistakes
⚠ Confusing angle of elevation with angle of depression
✗ Students write 'angle of elevation = 30°' when the problem clearly states 'angle of depression = 30°'.
→ Angle of elevation is measured upward from the horizontal; angle of depression is measured downward. Using the wrong one flips the triangle orientation and leads to incorrect side identification.
✓ Identify whether the observer is looking up (elevation) or down (depression). Draw the horizontal line from the observer's eye and mark the angle accordingly.
⚠ Forgetting to add the observer's height
✗ Students calculate AE = 28.5 m and write 'height of chimney = 28.5 m' without adding the observer's height of 1.5 m.
→ The observer's eye is not at ground level; the total height of the object includes the height from the ground to the observer's eye.
✓ Always add the observer's height (if given) to the calculated height from the triangle. In Example 3, AB = AE + BE = 28.5 + 1.5 = 30 m.
⚠ Using the wrong trigonometric ratio
✗ In a ladder problem, students use tan 60° = BD/DC instead of sin 60° = BD/BC.
→ When the unknown is the hypotenuse (ladder length), tan uses opposite and adjacent, not hypotenuse. Sin or cos should be used when the hypotenuse is involved.
✓ Identify which sides are known and which is unknown. If the hypotenuse is unknown and the opposite side is known, use sin. If adjacent is known, use cos.
⚠ Mixing up shadow lengths in two-angle problems
✗ Students set tan 60° = h/(x+40) and tan 30° = h/x, reversing the correct assignment.
→ A smaller angle of elevation (30°) gives a longer shadow. So the longer shadow (x+40) corresponds to the smaller angle (30°), not the larger one.
✓ Remember: lower sun altitude = longer shadow. So tan 30° = h/(longer shadow) and tan 60° = h/(shorter shadow).
⚠ Incorrectly applying alternate angles in depression problems
✗ Students assume the angle of depression equals the angle inside the triangle at the observer's eye without drawing the horizontal.
→ The angle of depression is measured from the horizontal, not from the vertical. The angle inside the triangle at the observer is the complement (90° - angle of depression) if the triangle is drawn incorrectly.
✓ Draw the horizontal line from the observer's eye. The angle of depression is between this horizontal and the line of sight. The angle inside the triangle at the observer is the same as the angle of depression only if the triangle's base is horizontal.
⚠ Forgetting to rationalize denominators
✗ Students leave answers like 10/√3 or 3/√3 without rationalizing.
→ CBSE expects simplified answers with rationalized denominators. Leaving √3 in the denominator is considered incomplete.
✓ Rationalize: 10/√3 = (10√3)/3, and 3/√3 = √3. Always multiply numerator and denominator by √3.
Compare & Contrast
Direction of line of sight relative to horizontal
Angle of elevation
Angle of depression
Angle of elevation is formed when the line of sight is above the horizontal (looking up), while angle of depression is formed when the line of sight is below the horizontal (looking down).
Ratio of sides in a right triangle
tan A = opposite/adjacent
cot A = adjacent/opposite
tan A uses opposite over adjacent, while cot A is the reciprocal (adjacent over opposite). Both can be used to solve the same problem, but choosing the correct one depends on which side is known and which is unknown.
Exam Focus
Marks pattern: This chapter typically appears as: 1 MCQ (1 mark), 1 Very Short Answer (1 mark), 1 Short Answer (3 marks), and 1 Long Answer (5 marks). Total: 10 marks.
Most asked: Finding height of a tower or building using angle of elevation · Finding distance between two objects using angles of depression · Problems involving two angles (e.g., flagstaff on building, shadow length changes) · Problems with observer's height included
Don't skip: Long answer questions (5 marks) involving two angles of elevation or depression, such as finding the height of a multi-storeyed building or the width of a river.; Questions that require setting up two equations and solving simultaneously (e.g., shadow problems).
One-Line Revision (read aloud before the exam)
  1. The line of sight is the line drawn from the eye of an observer to the point in the object viewed.
  2. The angle of elevation is the angle formed by the line of sight with the horizontal when the point is above the horizontal level.
  3. The angle of depression is the angle formed by the line of sight with the horizontal when the point is below the horizontal level.
  4. To find the height of a tower, use tan of the angle of elevation = height / distance from the foot.
  5. When the angle of elevation is 45°, the height of the object equals the horizontal distance from the observer.
  6. In ladder problems, use sin to find the length of the ladder when the height to be reached is known.
  7. In shadow problems, a smaller angle of elevation gives a longer shadow.
  8. Always add the observer's height to the calculated height to get the total height of the object.
  9. The angle of depression from a point equals the angle of elevation from the point being viewed (alternate interior angles).
  10. To find the width of a river from a bridge, add the distances to both banks calculated using tan of the angles of depression.
  11. In two-angle problems (e.g., flagstaff on a building), the distance from the point to the building is the same in both triangles.
  12. Rationalize denominators in final answers, e.g., 10/√3 = (10√3)/3.
  13. Draw a clear diagram for every problem — it helps identify which trigonometric ratio to use.
  14. The three pieces of information needed to find the height of a minar are: distance from the foot, angle of elevation, and height of the observer.

MCQs — 10 Questions with Answers

Q1. A surveyor needs to find the height of a building but can only measure the horizontal distance and observe angles. She stands at ground level 50 m away from the building and measures the angle of elevation to the top. Which trigonometric ratio should she use to find the height directly?

  • A. tan(angle of elevation) = height / horizontal distance ✓
  • B. sin(angle of elevation) = horizontal distance / height
  • C. cos(angle of elevation) = height / slant distance
  • D. cot(angle of elevation) = height / horizontal distance

Answer: A — tan(θ) = opposite/adjacent, where opposite is the height and adjacent is the horizontal distance; option B inverts the ratio and misidentifies the trigonometric function.

Q2. A person observes a bird flying at a height above them. The angle of elevation from their eye level to the bird is 35°. Later, the bird descends to a position below the level of the observer. Which angle should the observer now measure, and why?

  • A. Angle of elevation, because the bird is still visible from the same position
  • B. Angle of depression, because the bird is now below the horizontal level through the observer's eye ✓
  • C. The same angle of 35°, because the bird's actual distance has not changed
  • D. Angle of elevation measured downward, which is a special case of depression

Answer: B — Angle of depression is defined as the angle below the horizontal when looking down at an object; option A confuses position with the type of angle, and option C ignores the definition based on relative height.

Q3. Assertion (A): If the angle of elevation of the top of a tower from a point on the ground doubles, the height of the tower also doubles. Reason (R): The angle of elevation and height of an object are directly proportional because both are measured from the same reference point. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true ✓

Answer: D — A is false: doubling the angle (e.g., from 30° to 60°) does not double the height due to the nonlinear nature of trigonometric functions; R is true but not a valid explanation since angle and height are not directly proportional.

Q4. A ladder is placed against a vertical wall. An observer measures the angle of elevation from the foot of the ladder to the top of the ladder against the sky. Which statement about this angle is correct?

  • A. It is the angle between the ladder and the ground, which equals the angle of elevation ✓
  • B. It is not an angle of elevation because the top of the ladder is not higher than the observer's eye level
  • C. It is the angle between the ladder and the vertical wall, not an angle of elevation
  • D. It is the angle of elevation only if the observer stands at the foot of the ladder

Answer: A — The angle between the ladder and the ground is measured from the horizontal ground to the ladder, which is the definition of angle of elevation; option C confuses it with the angle between the ladder and wall.

Q5. Assertion (A): In a right triangle formed by angle of elevation problems, the horizontal distance is always the adjacent side to the angle of elevation. Reason (R): The angle of elevation is always measured at the observer's position on the ground against the horizontal line. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both statements are true and R correctly explains why the horizontal distance must be adjacent to the angle at the observer's position; the horizontal reference is fundamental to defining which side is adjacent.

Q6. A student claims: 'If I know the angle of elevation and the height of an object, I can always find the horizontal distance using trigonometry.' Under what condition is this claim false?

  • A. When the angle of elevation is 45°, because tan(45°) = 1 is a special case
  • B. When the height given is the vertical distance from the observer's eye level, not from the ground
  • C. The claim is always true; knowing angle and height is sufficient to find horizontal distance using tan(θ) = height/distance ✓
  • D. When the observer is not standing on level ground at the base of the object

Answer: C — The claim is always true mathematically; option D is a practical limitation but not a mathematical one, and option A incorrectly treats 45° as a special case for the method itself.

Q7. Assertion (A): The angle of elevation from point P to the top of a tower and the angle of depression from the top of the tower to point P are always equal. Reason (R): These angles are alternate interior angles formed by the line of sight and a horizontal line. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both are true: the angles are equal as alternate interior angles formed when a transversal (line of sight) cuts two parallel horizontal lines (one at P, one at the tower top); R correctly explains the geometric reason.

Q8. A surveyor standing at point A on a flat field observes two vertical poles. The angle of elevation to the top of pole B is 30°, and the horizontal distance is 60 m. For pole C at the same location but with angle of elevation 60° from A, the surveyor concludes pole C is twice the height of pole B. Is this reasoning correct?

  • A. Yes, because 60° is double 30°, so the height must double
  • B. No, because tan(60°) ÷ tan(30°) = √3, not 2, so pole C is √3 times taller ✓
  • C. Yes, the relationship between angle and height is always proportional
  • D. No, the heights cannot be compared without knowing the horizontal distances to each pole

Answer: B — Using tan(θ) = height/distance with the same distance: height ratio = tan(60°)/tan(30°) = √3/（1/√3) = 3, not 2; option A incorrectly assumes linear proportionality between angle and height.

Q9. Assertion (A): When finding the height of a building using angle of elevation, the observer's height must be added to the calculated height from the horizontal line through the observer's eyes. Reason (R): The angle of elevation is measured from the observer's eye level, not from the ground level. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — Both are true and R directly explains why A is correct: since the angle is measured from eye level, the perpendicular distance found represents height above eye level, requiring addition of observer's height to get total building height.

Q10. Two observers at different horizontal distances from a tower measure the same angle of elevation to its top. Which conclusion is correct?

  • A. They must be at the same point because the same angle means the same height and distance
  • B. The observer closer to the tower must be standing at a higher elevation than the other
  • C. This is impossible; different distances cannot yield the same angle of elevation
  • D. They cannot be at the same height, because different distances with the same angle require different eye levels ✓

Answer: D — For the same tower height and angle, tan(θ) = h/d means if distances differ, the heights above eye level must differ proportionally, so observers must be at different elevations; option B is correct in conclusion but option D is the most precisely reasoned answer.

Flashcards

What is the angle of elevation?

The angle formed by the line of sight with the horizontal when the point being viewed is above the horizontal level.

What is the angle of depression?

The angle formed by the line of sight with the horizontal when the point being viewed is below the horizontal level.

Formula: tan(angle) in height-distance problems

tan(angle) = height/horizontal distance (opposite/adjacent in right triangle).

When observer height is given, how do we find total height of object?

Total height = calculated height from eye level + observer's eye height.

In a shadow problem with two sun angles, what do we set up?

Two equations using tan(30°) and tan(60°) with two different shadow lengths differing by given amount.

How to find ladder length when angle of inclination is given?

Use sin(angle) = height to reach / ladder length, then solve for ladder length (hypotenuse).

In depression angle problems between two buildings, what triangles form?

Two right triangles sharing a common vertical height, one for top point and one for bottom point of the second building.

Why do we use tan ratio in most height-distance problems?

Because tan relates the two quantities we typically know or need: height (opposite) and horizontal distance (adjacent).

What does 'line of sight' mean in trigonometry applications?

The straight line drawn from the observer's eye to the point on the object being viewed.

In flagstaff problems, why are two separate right triangles used?

One triangle uses the building height to find distance; second triangle uses building plus flagstaff height to find flagstaff length.

Important Board Questions

Define angle of elevation and angle of depression with one example each. [2 marks]

State that angle of elevation is measured upward from horizontal when object is above eye level; angle of depression is measured downward when object is below eye level. Give one real-life example for each (e.g., looking up at plane for elevation, looking down from balcony for depression).

A ladder 10 m long leans against a wall making an angle of 30° with the ground. Find: (i) height reached on the wall (ii) distance of the ladder's foot from the wall. [3 marks]

Use sin 30° = height/10 to find height reached; use cos 30° = distance/10 to find distance from wall. Substitute sin 30° = 1/2 and cos 30° = √3/2 to get numerical answers.

From a point P on ground, the angle of elevation to the top of a 15 m tall building is 30°. A flag is hoisted on the building such that the angle of elevation to the flag from P becomes 45°. Find: (i) distance of building from P (ii) length of the flag. (Take √3 = 1.732) [5 marks]

First use tan 30° = 15/PA to find PA = 15√3 m. Then use tan 45° = (15 + flag)/PA to get flag = 15(√3 - 1) m. Show all algebraic steps and numerical substitution clearly.

Assertion-Reason Questions for Some Applications of Trigonometry

Q1. Assertion (A):

The angle of elevation is always measured from the horizontal upward to the line of sight when viewing an object above eye level.

Reason (R):

When an observer raises their head to look at an object above them, the line of sight makes an angle with the horizontal level that is called the angle of elevation.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true—the angle of elevation specifically refers to angles measured upward from horizontal when objects are above the horizontal line passing through the observer's eyes. The reason directly supports the assertion.

Q2. Assertion (A):

If a tower of height 10 m casts a shadow of 10 m at a given time, then the angle of elevation of the sun at that moment is 45°.

Reason (R):

The tangent of the angle of elevation equals the height of the object divided by the length of its shadow.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both assertion and reason are true. When height = shadow, tan θ = 1, which gives θ = 45°. However, the reason is not the correct explanation for why this specific angle results—it's more of a general formula statement rather than explaining the particular relationship that height = shadow gives 45°. The reason is true but not the direct cause of the assertion.

Q3. Assertion (A):

From the top of a building, if the angle of depression to a point on the ground is 60°, then this is the same as the angle of elevation from that ground point to the top of the building.

Reason (R):

The angle of depression and angle of elevation are complementary angles that sum to 90°.

Answer: C — A is true but R is false — The assertion is true—these angles are indeed equal as alternate interior angles when parallel horizontal lines are cut by a transversal (the line of sight). However, the reason is false. The angles of depression and elevation are not complementary; they are equal, and they don't necessarily sum to 90°. The reason provides an incorrect geometric relationship.

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Case-Based Questions (CBQ) for Some Applications of Trigonometry

Case Study:

A photographer wants to capture the height of a historical monument using angles of elevation. She stands 50 m away from the base of the monument and measures the angle of elevation to the top to be 60°. Later, she walks 30 m further back (now 80 m from the base) and measures the new angle of elevation to be 45°. Using these two measurements from different positions, she can verify the height of the monument using trigonometric calculations and ensure her measurements are consistent.

Q1. What is the height of the monument based on the first measurement (distance = 50 m, angle = 60°)?

  • A. 50√3 m ✓
  • B. 50/√3 m
  • C. 25√3 m
  • D. 100 m

Answer: A

Q2. What height does the second measurement suggest (distance = 80 m, angle = 45°)?

  • A. 40 m
  • B. 80 m ✓
  • C. 40√2 m
  • D. 80√3 m

Answer: B

Q3. Approximately, what is 50√3 m in meters? (Use √3 ≈ 1.73)

  • A. 76.5 m
  • B. 86.5 m ✓
  • C. 96.5 m
  • D. 106.5 m

Answer: B

Q4. Why might the two measurements give slightly different heights in practice?

  • A. The photographer measured from different heights above ground
  • B. The monument is leaning, so measurements from different angles show different heights
  • C. Rounding and measurement errors in angles and distances can accumulate ✓
  • D. The angle of elevation changes with time of day

Answer: C

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Previous Year Questions (PYQ) for Some Applications of Trigonometry

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 5 marks

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building and distance between the building and the tower. (Use √3 = 1.73)

CBSE 2026 · Set 31/4/2 · 5 marks

A statue, 2 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the bottom of the statue is 45°. Find the height of the pedestal and its distance from the point of observation on the ground. (use √3 = 1.73)

CBSE 2026 · Set 31/4/3 · 5 marks

A straight road leads to foot of a tower whose shadow is found to be 40 m longer when sun's altitude is 30° than when it is 60°. Find the height of the tower and length of shadow in both the situations. (Use √3 = 1.73)

CBSE 2026 · Set 31/5/1 · 1 mark

The length of a pendulum is 70 cm and it describes an arc of length 88 cm when swings. The angle subtended by the arc at the centre is

  • 36°
  • 70°
  • 72°
  • 80°

CBSE 2026 · Set 31/5/1 · 5 marks

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distance of the point from the poles.

CBSE 2026 · Set 31/5/2 · 3 marks

Chord AB of a circle subtends an angle of 120º at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

CBSE 2026 · Set 31/5/2 · 5 marks

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60º and 30º respectively. Find the height of the poles and the distance of the point from the poles.

CBSE 2026 · Set 31/1164/30 · 5 marks

Two poles of equal heights are standing opposite to each other on either side of the road which is 90 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 30° and 60° respectively. Find the height of the poles and the distances of the point from the poles. [Use √3 = 1.732]

CBSE 2026 · Set 31/1167/430 · 5 marks

The angle of elevation of the top of a tower as observed from a point in a horizontal plane through the foot of the tower is 30°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 60°. Find the height of the tower and the distance of first position from the tower.

CBSE 2026 · Set 31/5/1 · 1 mark

Elevated water storage tanks are built to store and supply water to nearby colonies. In the diagram given above, AB is an elevated water tank and CD is a nearby multistorey building. The building is 54 metres away from the water tank. From a window (W) of the building, the angle of elevation of top of the tank is 45° and angle of depression of its foot is 30°.

Write a relation between d (the height of window) and y.

CBSE 2026 · Set 31/1/1 · 1 mark

An injured bird was found on the roof of a building. The building is 15 m high. A fireman was called to rescue the bird. The fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of 60° with the ground in order to reach the roof.

Find the length of the ladder used by the fireman to reach the roof.

CBSE 2026 · Set 31/1/2 · 1 mark

From the figures given below, which of the following is true about the measure of P ?

  • P = 60°
  • P = 80°
  • P = 40°
  • The measure of P cannot be determined

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Class 10 Maths Formula Sheet

Every formula from all 14 chapters on one page, checked against the NCERT textbook and the CBSE 2026-27 syllabus. Free PDF download.

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Class 10 Mathematics Previous Year Questions, Chapter-wise

All 1107 CBSE board questions from 2022–2026, sorted by chapter and question type. Free PDF download.

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