Chapter 8 Introduction to Trigonometry

Case Study 1: Shadow of a Tree

Case Description:
In a park, a tall tree casts a shadow on the ground. At a certain time of the day, the height of the tree is measured to be 15 meters, and the length of the shadow is found to be 10 meters. To find the angle of elevation of the sun, the gardener uses trigonometric ratios. He calculates the angle using the relationship between the height of the tree and the length of the shadow. The gardener aims to determine the sun's position for planning future planting.

MCQs:

1. What is the ratio used to find the angle of elevation in this case?

   • A) $\tan \theta = \frac{\text{height}}{\text{shadow}}$
   • B) $\sin \theta = \frac{\text{height}}{\text{hypotenuse}}$
   • C) $\cos \theta = \frac{\text{shadow}}{\text{hypotenuse}}$
   • D) $\tan \theta = \frac{\text{shadow}}{\text{height}}$

2. If the angle of elevation is calculated to be $\theta$, which trigonometric function is used?

   • A) $\sin$
   • B) $\cos$
   • C) $\tan$
   • D) All of the above

3. What is the angle of elevation of the sun if the height of the tree is 15 meters and the shadow is 10 meters?

   • A) 30°
   • B) 36.87°
   • C) 45°
   • D) 53.13°

4. If the height of the tree increases to 20 meters while keeping the shadow length constant at 10 meters, what will be the new angle of elevation?

   • A) 30°
   • B) 53.13°
   • C) 60°
   • D) 75°

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Case Study 2: The Ladder Problem

Case Description:
A ladder is leaning against a wall, making contact at a height of 8 meters from the ground. The foot of the ladder is 6 meters away from the wall. The contractor wants to ensure that the ladder is placed at a safe angle to prevent slipping. To find this angle, he uses trigonometric ratios to calculate the angle between the ground and the ladder.

MCQs:

1. What is the height at which the ladder touches the wall?

   • A) 6 meters
   • B) 8 meters
   • C) 10 meters
   • D) 12 meters

2. Which trigonometric ratio can be used to find the angle $\theta$ formed by the ladder with the ground?

   • A) $\sin \theta$
   • B) $\cos \theta$
   • C) $\tan \theta$
   • D) $\sec \theta$

3. What is the angle $\theta$ formed by the ladder with the ground?

   • A) 30°
   • B) 45°
   • C) 53.13°
   • D) 60°

4. If the height at which the ladder touches the wall increases to 10 meters, while the distance from the wall remains the same, what will be the new angle $\theta$?

   • A) 36.87°
   • B) 45°
   • C) 53.13°
   • D) 60°

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Case Study 3: The Kite in the Sky

Case Description:
A kite is flying at a height of 20 meters. The angle of elevation from a point on the ground 15 meters away from the base of the kite is measured. To determine the angle of elevation, the observer uses trigonometric ratios to analyze the situation. This information helps in understanding how high kites can be flown safely in the vicinity.

MCQs:

1. What is the height of the kite?

   • A) 15 meters
   • B) 20 meters
   • C) 25 meters
   • D) 30 meters

2. What is the distance from the observer to the base of the kite?

   • A) 10 meters
   • B) 15 meters
   • C) 20 meters
   • D) 25 meters

3. Which trigonometric ratio can be used to find the angle of elevation $\theta$?

   • A) $\sin \theta$
   • B) $\tan \theta$
   • C) $\cos \theta$
   • D) All of the above

4. If the angle of elevation is $\theta$, what is the value of $\tan \theta$ based on the given measurements?

   • A) $\frac{4}{3}$
   • B) $\frac{3}{4}$
   • C) $\frac{5}{3}$
   • D) $\frac{20}{15}$

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Case Study 4: The Viewing Tower

Case Description:
A viewing tower stands at a height of 30 meters. From a point on the ground, an observer measures the angle of elevation to the top of the tower. Using trigonometric principles, the observer can find the angle of elevation and decide on the best position to view the surrounding area. This is crucial for planning outdoor events.

MCQs:

1. What is the height of the viewing tower?

   • A) 20 meters
   • B) 25 meters
   • C) 30 meters
   • D) 35 meters

2. If the observer is standing 40 meters away from the base of the tower, which trigonometric ratio helps find the angle of elevation?

   • A) $\sin$
   • B) $\cos$
   • C) $\tan$
   • D) All of the above

3. What is the angle of elevation if the height of the tower is 30 meters and the distance from the observer to the tower is 40 meters?

   • A) 36.87°
   • B) 45°
   • C) 53.13°
   • D) 60°

4. If the observer moves back to 50 meters from the tower, what will the new angle of elevation be?

   • A) 30°
   • B) 36.87°
   • C) 45°
   • D) 53.13°

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Case Study 5: The Airplane Takeoff

Case Description:
An airplane takes off and ascends at an angle of elevation of 30° to reach a height of 1000 meters. Observers on the ground wish to find out how far the airplane is from them horizontally at this height. They use trigonometric ratios to calculate the horizontal distance, which is important for flight path planning and safety measures.

MCQs:

1. What is the angle of elevation of the airplane during takeoff?

   • A) 15°
   • B) 30°
   • C) 45°
   • D) 60°

2. What is the height of the airplane during takeoff?

   • A) 500 meters
   • B) 1000 meters
   • C) 1500 meters
   • D) 2000 meters

3. Which trigonometric function helps calculate the horizontal distance from the observers to the airplane?

   • A) $\sin$
   • B) $\cos$
   • C) $\tan$
   • D) All of the above

4. What is the horizontal distance of the airplane from the observers if it is 1000 meters high and ascending at an angle of elevation of 30°?

   • A) 1000 meters
   • B) 1155 meters
   • C) 1732 meters
   • D) 2000 meters