Some Applications of Trigonometry – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

Trigonometry finds heights and distances that cannot be measured directly, using angles of elevation and depression.

Angles

• Angle of elevation: angle above the horizontal when looking up at an object.
• Angle of depression: angle below the horizontal when looking down at an object.
• Angle of elevation from A to B equals the angle of depression from B to A (alternate angles).

Method

• Draw a right triangle with the height as the perpendicular and the distance as the base.
• Use tan θ = height/distance most often.
• Use sin or cos when the line of sight (hypotenuse) is involved.

Key terms

Line of sight

the line from the observer's eye to the object

Angle of elevation

angle between the horizontal and the line of sight to a higher object

Angle of depression

angle between the horizontal and the line of sight to a lower object

Formulas

• tan θ = height / horizontal distance
• Values: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3

Important questions with answers

Quick revision

• Elevation looks up; depression looks down.
• Most problems need tan.
• Draw the figure first — marks are given for it.

Exam tips

• Add the observer's height when the angle is measured from the eye level, not the ground.
• Use √3 ≈ 1.732 only at the final step.

Practice MCQs

1. The angle of elevation of the sun when a pole's shadow equals its height is

   1. 30°
   2. 45°
   3. 60°
   4. 90°
   Show answer

   Answer: (b) 45°. tan θ = 1 ⇒ θ = 45°.

2. Looking down at a boat from a cliff, the angle formed is called

   1. elevation
   2. reflex
   3. depression
   4. right
   Show answer

   Answer: (c) depression. It is below the horizontal.

3. A tower 30 m high is seen at 45° from a point. The point is how far from the foot?

   1. 10√3 m
   2. 15 m
   3. 30√3 m
   4. 30 m
   Show answer

   Answer: (d) 30 m. tan 45° = 30/d ⇒ d = 30 m.

4. A kite string 100 m long makes 60° with the ground. Height of the kite is

   1. 50 m
   2. 50√3 m
   3. 100√3 m
   4. 100 m
   Show answer

   Answer: (b) 50√3 m. sin 60° = h/100 ⇒ h = 50√3 m.

5. A tower subtends 30° at a point 30 m from its foot. Its height is

   1. 10√3 m
   2. 30√3 m
   3. 15 m
   4. 10 m
   Show answer

   Answer: (a) 10√3 m. h = 30 × tan 30° = 30/√3 = 10√3 m.

6. Angle of elevation of a tower top from 20√3 m away is 30°. Height is

   1. 10 m
   2. 60 m
   3. 20√3 m
   4. 20 m
   Show answer

   Answer: (d) 20 m. h = 20√3 × 1/√3 = 20 m.

7. Shadow of a tower is 40 m longer when the sun's altitude changes from 60° to 30°. The height is

   1. 40 m
   2. 40√3 m
   3. 20 m
   4. 20√3 m
   Show answer

   Answer: (d) 20√3 m. h(√3 − 1/√3) = 40 ⇒ h × 2/√3 = 40 ⇒ h = 20√3 m.

8. From the top of a 75 m lighthouse, angles of depression of two ships on the same side are 30° and 45°. Distance between the ships is

   1. 75(√3 − 1) m
   2. 75√3 m
   3. 75 m
   4. 75(√3 + 1) m
   Show answer

   Answer: (a) 75(√3 − 1) m. Distances 75√3 and 75, difference 75(√3 − 1) m.