Some Applications of Trigonometry — Class 10 MCQs with Answers

Some Applications of Trigonometry — important questions & MCQs with answers (Class 10 Mathematics)

6 solved questions from this chapter's difficulty levels, each with its answer and explanation. Tap an option to check yourself.

0 of 6 answered

1. Q1 of 6Easy

   Angle of elevation is measured from:

   AVertical, downwardBHorizontal, downward to objectCHorizontal, upward to objectDVertical, upward to object
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   Answer: Horizontal, upward to object

   Explanation: Angle of elevation is the angle a horizontal line of sight makes turning upward to sight an object above it — measuring from the vertical, not the horizontal, is the usual mix-up.

2. Q2 of 6Easy

   tan 45° =

   A0B1/√3C√3D1
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   Answer: 1

   Explanation: In a right triangle with a 45° angle, opposite and adjacent sides are equal, so tan 45° = opposite/adjacent = 1 — not √3 (tan 60°) or 1/√3 (tan 30°).

3. Q3 of 6Medium

   From two points 30 m apart on the same side of a tower, the angles of elevation of the top of the tower are 30° and 60°. The height of the tower is:

   

   A15 mB30 mC15√3 mD30√3 m
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   Answer: 15√3 m

   Explanation: Let height h, distance from foot to closer point d. tan 60° = h/d ⇒ d = h/√3. tan 30° = h/(d + 30) ⇒ (d + 30) = h√3. So h/√3 + 30 = h√3 ⇒ 30 = h√3 − h/√3 = (2h/√3) ⇒ h = 15√3.

4. Q4 of 6Medium

   Assertion (A): From a point on the ground 15 m from the foot of a vertical tower, the angle of elevation of its top is 60°. So the tower is 15√3 m high. Reason (R): The angle of depression is formed when the point being viewed is below the horizontal level.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true but R is not the correct explanation of A

   Explanation: Both are true. A follows from tan 60° = height ÷ 15, so height = 15√3 m. R is a correct definition, but this problem uses an angle of elevation, so R does not explain A.

5. Q5 of 6Hard

   Cliff is 100 m high. Boat at 45° depression is at distance:

   

   A50 mB100√2 mC100 mD100/√3 m
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   Answer: 100 m

   Explanation: By alternate angles the boat's elevation angle also is 45°, so tan 45° = 100/d = 1, giving d = 100 m — the height-equals-distance shortcut at 45°.

6. Q6 of 6Hard

   Assertion (A): The shadow of a vertical tower is 40 m longer when the Sun's altitude is 30° than when it is 60°. So the tower is 20√3 m high. Reason (R): If h m is the height and x m is the shorter shadow, then tan 60° gives h = √3 x and tan 30° gives h = (x + 40)/√3.

   ABoth A and R are true and R is the correct explanation of ABBoth A and R are true but R is not the correct explanation of ACA is true but R is falseDA is false but R is true
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   Answer: Both A and R are true and R is the correct explanation of A

   Explanation: Both are true. Solving h = √3 x and h = (x + 40)/√3 gives 3x = x + 40, so x = 20 and h = 20√3 m. R gives exactly the working that leads to A, so it explains A.

Some Applications of Trigonometry notes: key concepts (Class 10 Mathematics)

Trigonometry measures what cannot be reached: the height of a tower, the width of a river, the altitude of a balloon. This chapter formalises the angle of elevation and depression and uses the standard-angle ratios to solve real height-and-distance problems.

Line of sight

The straight line from the observer's eye to the object being viewed; every elevation and depression angle is measured from it.

Horizontal line

The reference line through the observer's eye, parallel to the ground, against which the line of sight is measured.

Angle of elevation

The angle between the horizontal and the line of sight when the object is ABOVE the observer's eye level.

Angle of depression

The angle between the horizontal and the line of sight when the object is BELOW the observer's eye level.

Elevation equals depression

The angle of depression from a tower top to a point equals the angle of elevation from that point to the top, being alternate angles.

Choosing the right ratio

Pick the ratio linking the side you know to the side you want — tan for two legs, sin or cos when the hypotenuse is involved.

Height of an object

Modelled as the vertical side of a right triangle whose horizontal side is the distance from the observer.

Observer's eye height

When the observer's height is given it must be ADDED to the height computed from the triangle, since the triangle starts at eye level.

Two-triangle problems

Two observation points give two equations sharing an unknown; subtracting or dividing them eliminates it.

Standard-angle values in context

Problems are set at 30°, 45° and 60° so exact surds appear; 45° in particular makes the height equal the distance.