# Heights and Distances

Trigonometry is the study of the relationship between the length of sides and angles of a triangle. A triangle is a closed shape consisting of three sides. The relation between the heights and distances of objects can be understood using trigonometry. We will learn about the same in this topic.

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## Trigonometry

Trigonometry has found its importance in multiple fields. It has found its applications in the fields ranging from Engineering to Architecture to Astronomy as well. Trigonometry can be used in these fields by to measure distances and angles by assuming lines that connect the points.

## Heights and Distances

Height is the measurement of an object in the vertical direction and distance is the measurement of an object from a particular point in the horizontal direction. If we imagine a line connecting the point of observation to the topmost point of the object then the horizontal line, vertical line and the imaginary line will form a triangle.

Observe the figure. Consider the observer to be at point C. The height of the object is shown by line AB. The distance of the object from the observer is given by line BC. The object may or may not be perpendicular to the ground. Line AC represents the Line of Sight when the observer is observing the topmost point of the object. Angle α represents the angle of elevation and Angle β represents the angle of depression.

Using trigonometry, if we are provided with any of the two quantities that may be a side or an angle, we can calculate all the rest of the quantities. By the law of alternate angles, the angle of elevation and angle of depression are consequently equal in magnitude (α = β). Tan α is equal to the ratio of the height and distance.

You can be provided with any two of the following information:

• The distance of the object from the observer
• The height of the object
• Angle at which the observer views the topmost point of the object (angle of elevation)
• The angle at which the observer views the object when the observer is on top of a tower/building (angle of depression)

and you can calculate the rest. Here are some examples for you in the following section.

## Solved Examples for You on Heights & Distances

Question 1: An observer 1.5 m tall is 28.5 m away from a tower. The angle of elevation of the top of the tower from his/her eyes has measure 45. What is the height of the tower?

1. 28.5 m
2. 30 m
3. 27 m
4. 1.5 m

We can form the above figure by given data,
BD=CE=1.5

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$

In ΔABC,CBA=90º
tan45º=AB/BC
1=AB/28.5
AB=28.5

$\mathrm{}\mathrm{}{}^{}$$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$

Now, h=AB+BD=28.5+1.5=30
Hence, the height of the tower is 30 m

Question 2: What does height mean?

Answer: Basically, height means altitude or elevation that refers to the distance above a level. Furthermore, height denotes extent upward (as from foot to head) as well as any measurable distance above a given level. For example, the height of a tree, human, mountain, tower, etc.

Question 3: What is the difference between height and hight?

Answer: The major difference between height and hight is that height refers to the distance between the highest and lowest end of an object. On the other hand, hight refers to a mountain located in the United States of America.

Question 4: How do you find the height of objects in math?

Answer: In mathematics, you can calculate the height of an object using the distance and angles. Here distance is the horizontal distance between the object and the angle is the angle above the horizontal of the top of the object, which gives the height of the object.

Question 5: What is the maximum height?

Answer: It refers to the highest vertical position along the path of its trajectory. Moreover, the maximum height of the projectile depends upon the velocity and angle of the launch of the object and the acceleration due to gravity.



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##### Some Applications of Trigonometry
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