Trigonometry (Applications Case Study) (Class 10 Mathematics)
Real-world applications of trigonometry involving heights, distances, angles of elevation and depression. CBSE-style case questions on surveying, engineeri
TL;DR: Real-world applications of trigonometry involving heights, distances, angles of elevation and depression. CBSE-style case questions on surveying, engi…
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Real-world applications of trigonometry involving heights, distances, angles of elevation and depression. CBSE-style case questions on surveying, engineeri
Trigonometry (Applications Case Study) MCQs with Answers & Explanations
Q1. A boy stands 30 meters from the base of a tree and observes that the angle of elevation to the top is 45 degrees. What is the height of the tree?
- 30 sqrt(2) m
- 60 m
- 30 m ✓ (correct)
- 15 m
Explanation: Using tan(45) = height/distance, we have 1 = h/30, so h = 30 m. At 45 degrees, height equals horizontal distance.
Q2. From the top of a building 80 meters tall, the angle of depression to a car on the ground is 30 degrees. How far is the car from the base of the building?
- 40 sqrt(3) m
- 160 m
- 40 m
- 80 sqrt(3) m ✓ (correct)
Explanation: tan(30) = height/distance, so 1/sqrt(3) = 80/d, giving d = 80*sqrt(3) meters.
Q3. A ladder leans against a wall at an angle of 60 degrees with the ground. If the ladder is 10 meters long, what height on the wall does it reach?
- 5 sqrt(3) m ✓ (correct)
- 5 m
- 10 sqrt(3) m
- 10 m
Explanation: sin(60) = height/hypotenuse, so sqrt(3)/2 = h/10, giving h = 5*sqrt(3) meters.
Q4. A surveyor measures an angle of 60 degrees from point A to the top of a tower 100 meters away horizontally. Later, moving 50 meters closer (point B), the angle is now 75 degrees. This scenario best demonstrates which concept?
- Complementary angles
- Angle of elevation changes with distance ✓ (correct)
- Inverse trigonometric functions
- Law of Sines
Explanation: As the observer moves closer to the tower, the angle of elevation increases from 60 to 75 degrees, showing the inverse relationship between distance and angle of elevation.
Q5. In a right triangle, sin(A) = 3/5. What is cos(A)?
- 4/3
- 5/4
- 4/5 ✓ (correct)
- 3/4
Explanation: If sin(A) = 3/5, then opposite = 3 and hypotenuse = 5. Using Pythagoras: adjacent = sqrt(25-9) = 4. So cos(A) = 4/5.
Q6. Two poles of heights 10 m and 20 m are 15 m apart. What is the angle of elevation from the top of the shorter pole to the top of the taller pole?
- 30 degrees
- 45 degrees
- 60 degrees
- tan^-1(2/3) ✓ (correct)
Explanation: The vertical difference is 20-10 = 10 m. Horizontal distance is 15 m. tan(angle) = 10/15 = 2/3. angle = tan^-1(2/3).
Frequently Asked Questions
What is the relationship between angle of elevation and angle of depression?
Angle of elevation is the angle above the horizontal when looking up at an object from a lower position. Angle of depression is the angle below the horizontal when looking down at an object from a higher position. For an observer at height h and object at horizontal distance d, the angle of elevation from the object to the observer equals the angle of depression from the observer to the object (alternate interior angles with respect to the horizontal).
How do we choose between sin, cos, and tan in trigonometry problems?
Identify which sides and angles are given and which are unknown. Use SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Choose the ratio containing the known angle, known side, and unknown side you need to find. In practical problems, first determine if you have the angle and can find opposite, adjacent, or hypotenuse.
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