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Chapter 9: Problem 2

VOCABULARY Compare an angle of elevation to an angle of depression.

Short Answer

Expert verified
Both 'angle of elevation' and 'angle of depression' are angles made with the horizontal, the former when looking upwards and the latter when looking downwards. Despite different contexts, they are congruent when considered vertically opposite angles with the observer's eye as the vertex.

Step by step solution

01

Definition of Angle of Elevation

The angle of elevation is defined as the angle formed by the line of sight (when looking upwards) with the horizontal line. Suppose a person is standing on the ground and looking up at an object in the sky. The angle the person's line of sight makes with the ground (or the horizontal) is termed as the angle of elevation.
02

Definition of Angle of Depression

On the other hand, the angle of depression is the angle formed by the line of sight when an observer looks downward with the horizontal line. Suppose a person is standing on top of a hill and looking downwards at an object on the ground. The angle formed by the person's line of sight (downwards) and the horizontal is called the angle of depression.
03

Comparing Angle of Elevation and Depression

Now, comparing these two, they are vertically opposite angles when the observer's eye is considered as the vertex, and thus they're congruent. In simpler words, if an observer moves their line of sight from a higher object down to a lower object, the angle of elevation from the lower object is equal to the angle of depression from the higher object.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angle of Elevation
Imagine you're standing on the ground and gazing up at a high-flying kite. The angle between your sight line to the kite and the horizontal ground is called the angle of elevation. This concept helps us describe the steepness of the gaze as you look upwards. In mathematical terms, if you know the distance to the base of the object and the height you are looking up to, you can use the tangent function of trigonometry.
  • The angle of elevation (胃) can be found using: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
  • 'Opposite' is the height of the object, and 'adjacent' is the horizontal distance to the object.
This concept is widely used in real-world applications like determining the height of buildings and navigation for airplanes.
Angle of Depression
Angles of depression come into play when you're situated at a higher point and looking downwards, like being on a balcony looking at a kid playing in the park below. This angle is formed between your sight line looking downwards and the horizontal.To measure the angle of depression, one can consider the triangle formed where:
  • The line of sight forms the hypotenuse.
  • The horizontal line serves as the adjacent side.
  • When you know the distance from the base of your standing point to the object and the height you're looking down from, you can calculate it using the tangent function as: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)
One interesting fact about the angle of depression is that it's congruent to the angle of elevation when viewed from the object looking up.
Geometry Concepts
Geometry helps us explore the relationships between angles, lines, and surfaces, linking concepts like angles of elevation and depression beautifully. These angles rely on understanding right triangles and how to solve them using trigonometric functions, particularly tangent. Within a right triangle:
  • The primary focus is on understanding right angles.
  • Trigonometric functions such as sine, cosine, and tangent help relate different sides of the triangle to the angles.
  • Using these fundamental principles, you can also apply them to solve real-world problems, such as determining heights and distances.
By grasping these geometry concepts, you bridge the gap between basic trigonometry and its application in day-to-day scenarios, making complex calculations easy and intuitive.

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Most popular questions from this chapter

The latitude of a point on Earth is the degree measure of the shortest arc from that point to the equator. For example, the latitude of point \(P\) in the diagram equals the degree measure of \(\operatorname{arc} P E\). At what latitude \(\theta\) is the circumference of the circle of latitude at \(P\) half the distance around the equator?

MODELING WITH MATHEMATICS You are standing 120 feet from the base of a 260-foot building. You watch your friend go down the side of the building in a glass elevator. a. Write an equation that gives the distance \(d\) (in feet) your friend is from the top of the building as a function of the angle of elevation \(\theta\). b. Graph the function found in part (a). Explain how the graph relates to this situation.

Given an angle \(\theta\) in standard position with its terminal side in Quadrant III, explain how you can use a reference angle to find \(\cos \theta\).

Your friend states that for functions of the form \(y=a \sin b x\) and \(y=a \cos b x\), the values of \(a\) and \(b\) affect the \(x\)-intercepts of the graph of the function. Is your friend correct? Explain.

Write an expression in terms of the integer \(n\) that represents all the \(x\)-intercepts of the graph of the function \(y=\cos 2 x\). Justify your answer.
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