Saturday, July 13, 2024

# How To Find Angle Of Elevation And Depression

## Angle Of Elevation And Depression Comparison

Angle of Elevation and Depression Word Problems Trigonometry, Finding Sides, Angles, Right Triangles

The angle of depression is just the opposite scenario of the angle of elevation. In this case, the observer is standing at height and the object is kept below the line of sight of the observer. We can define it as if the object is kept below the eye level of the observer, then the angles formed between the horizontal line and the observers line of sight is called the angle of depression.

The formula of the angle formed here is given by;

tangent of angle of depression = Opposite side/adjacent side

## Angle Of Depression Definition Formulas Examples

The angle of depression is created when the observer is higher than the object he is looking at. If a person looks at an object that is located at a distance lower than the person, the angle is formed below the horizontal line drawn with the level of the eye of the person and line joining object with the persons eye. This angle is calculated by using the concept of trigonometry. Get the definition, formulas, and example questions with answers in the below sections.

## What Is An Angle Of Depression

In contrast to the angle of elevation, the angle of depression is the angle at which you would draw a line from a point on a horizontal line to intersect another point that falls below the line. Using the x-axis example from before, the angle of depression would require you to choose a point on the x-axis and draw a line from it to another point that was somewhere below the x-axis. The angle of that line in comparison to the x-axis itself would be the angle of depression. In the bird scenario, imagine the bird itself flying along an imaginary horizontal plane. The angle that the bird would look along to look down and see you standing on the ground would be the angle of depression.

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## Angle Of Depression Formula

With angles of elevation, if two of the sides of the right triangle are known, then the formula for the angle of depression is given as below:

= tan-1

See the below diagram, where is the angle of inclination, such as,

ABO = Angle of elevation

Hence, ABO = O =

## What Is Angle Of Elevation And Depression

The angle of elevation is the angle between a horizontalline and the line joining the observers eye to some object above the horizontal line.

The angle of depression is the angle between a horizontalline and the line joining the observers eye to some object beneath the horizontal line.

In real world situations, we often discuss the angles of elevation and depression.The angles of elevation and depression is used often in word problems, especially thoseinvolving a persons line of sight as they look up at an object.

This video will explain what is the angle of elevation and what is the angle of depression.It will also give some examples of how to use the angles of elevation and angles of depression.

Angle of Elevation/Depression Story Problems

Examples:The angle of elevation from point A to the top of a cliff is 38 degrees. If point A is 80feet from the base of the cliff, how high is the cliff? Let x be the height of the cliff.

• From the top of the tower, the angle of depression to a stake on the ground is 72 degrees.The top of the tower is 80 feet above the ground. How far is the stake from the foot of the tower?
• A tree 40 feet high casts a shadow 58 feet long. Find the measure of elevation of the sun.
• A ladder leaning against a house makes an angle of 60 degrees with the ground. The foot of theladder is 7 feet from the foundation of the house. How long is the ladder?
• Angles of Elevation and Depression

Examples:

Examples:

Examples:

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## Find The Angle Of Elevation

The angle framed by the line of sight and the horizontal is known as angle of elevation. It can be estimated from the known values of height and distance of the object. In other words, Angles of elevation or inclination are angles above the horizontal. Like looking up from ground level towards the top of a flagpole. Use this online calculator to find the angle of elevation by entering the values of height and distance of the object.

Where,

#### Example

An aeroplane flies at a height of 100mm, the distance of the plane from the observers point of view is 123mm.Angle of Elevation = atan = 39.11 degrees.

## Trigonometry Problems Involving Angle Of Depression

Angle of Depression :

The angle of depression is an angle formed by the line of sight with the horizontal when the point is below the horizontal level. That is, the case when we lower our head to look at the point being viewed.

To find questions 1 to 3, please visit the page Trigonometry Word Problems with Angle of Depression”

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## How To Find The Angle Of Elevation & Depression

Animals and artists, pilots and pelicans all instinctively use the angle of depression. From a park ranger in a fire tower to that pilot taxiing or landing an A380 airliner, the angle of depression provides important information about distance and height.

Its opposite, the angle of elevation, is what humans do when they want to look up at tall mountains, nests in trees, or towering buildings.

## Angle Of Elevation Definition

Learn how to determine the angle of elevation and angle of depression

The angle of elevation is an angle that is formed between the horizontal line and the line of sight. If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.

In the above figure, you can see, an observer is looking at the object, standing on the ground, forming an angle with the line of sight and horizontal line. Here, if we join an imaginary line between the object and end of the horizontal line, a right angle triangle is formed. Thus we can use here trigonometry concept to find the distance of the observer from the tower or building. Height of the tower or building or the height at which the object is kept will be considered as perpendicular and the horizontal line will be considered as adjacent side of the triangle formed. The related terminology are given here.

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## What Is The Angle Of Depression Angle Of Depression Definition

The simplest angle of depression definition is that it is the angle between the horizontal and the part of a line that is below the horizontal. In the image below, as the cat looks downwards towards point A, it creates a certain angle of depression from the horizontal.

Sometimes we may also see the angle of depression used to express the slope of a surface such as a mountainside, or a roadway. However, for such information we typically use the counterpart of the angle of depression – the elevation grade.

## Lesson Explainer: Angles Of Elevation And Depression Mathematics

In this explainer, we will learn how to solve real-world problems that involve angles of elevation and depression.

Before you start with this explainer, you should be confident finding angle measures and missing sides using right triangle trigonometry and the laws of sines and cosines.

Now, prior to looking at examples and recalling trigonometric ratios and the laws of sines and cosines, we will define what angles of elevation and depression are.

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## Sample Calculation Of Finding The Angle Of Depression

Time for an angle of depression example. Suppose we need to find the angle of depression, , of the boy’s line of sight from the top of the slide towards the girl at the bottom of the slide, as shown in the illustration below. We can see that the horizontal distance between the children is 3.0 meters and the vertical distance between their line of sights is 1.5 meters.

Using the angle of depression formula, we calculate angle of depression as follows:

= arctan

= arctan

= arctan

= 26.56505118°26.565°

From the calculation above, we can now say that the angle of depression, , is around 26.565° from the horizontal.

## Angle Of Depression Problems

Example 1:

The angles of elevation and depression of the top and bottom of a lamp post from the top of a 66 m high apartment are 60° and 30° respectively. Find The height of the lamp post. The difference between the height of the lamp post and the apartment. The distance between the lamp post and the apartment.

Solution:

Triangle AED forms a right triangle

So, tan 60° = \

3 = \

In trinagle ABC,

tan 30° = \

\ = \

BC = 663 –

\ = 663

ED = 663

Height of lamp post = ED + DC

= 198 + 66

The difference between height of the lamp post and the apartment

= 364 66

The distance between the lamp post and the apartment

BC = 663

= 114.31 m

Example 2:

An airplane is flying at a height of 2 miles above level ground. The angle of depression from the plane to the foot of the tree is 15°. What is the distance the plane must fly to be directly above the tree?

Solution:

To find the distance BA use the tangent function

tan 15° = \

0.26794919243 = \

So, the plane must fly 7.464 ft horizontally to be directly over the tree.

Example 3:

A buoy in the ocean is observed from the top of a 40-meter-high oil rig. The angle of depression from the top of the tower to the buoy is 6°. How far is the buoy from the base of the oil rig?

Solution:

The angle of depression from the top of the tower to the buoy = 6°

A buoy in the ocean is observed from the top of a 40-meter-high oil rig.

Tan 6° = \

h = \

h = \

It is approximately 380.6 m from the buoy to the base of the oil rig.

Example 4:

Solution:

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## Angles Of Elevation & Depression

• Example Problems
• If you have ever watched a commercial airplane roll back from an airport gate, you may have noticed a crewmember on the ground giving signals to the pilot in the cockpit, many feet above. Have you ever thought that the angle of sight for the ground crewmember is the same angle of sight for the pilot? The pilot’s viewing angle is called the angle of depression, while the ground crewmember’s viewing angle is the angle of elevation.

## Angle Of Elevation And Angle Of Depression

The angle of elevation and angle of depression are opposite to each other. The elevation angle is formed when it is between the line of sight and the horizontal line. And if the line of sight is above the horizontal line, then the angle is called the angle of elevation. In the angle of depression, the line of sight is downwards to the horizontal line.

ABO = O =

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## Angle Of Depression Explanation & Examples

When you look at an item below you, you can easily measure the angle of depression formed by your line of sight with the horizontal line. Just imagine you are standing at the top of the Pisa Tower and looking at an infinite horizon to enjoy the beautiful weather on a great rainy day. Suddenly your friend, on the ground, accidentally finds you and screams to say Hi. You;lower;your eyes to look to see your friend. You must realize that you created a certain angle as you look downwards toward your friend. This angle is called the angle of depression.;

The angle of depression is basically the measure of an angle between the horizontal line and line of sight of a persons eyes to any item below.The angle of elevation depends upon the movement of your eyes.;

After this lesson, we expect you to learn the concepts of the angle of depression and be able to confidently answer the following questions:;

• What is an angle of depression?
• How to find the angle of depression?
• How can we solve real-world problems using the angle of depression?

What Is an Angle of Depression?

When an observer is looking below at an object, the angle established by the line of sight with the horizontal line is called the;angle of depression.

Now, if the man is looking at the base of the wall, what should be the line of sight?

Looking at Figure 11-2, the angle $\theta$;represents the angle of depression.

## Angle Of Depression Definition

Angles Of Elevation And Depression – VividMath.com

The angle of depression is the angle between the horizontal line and the observation of the object from the horizontal line. It is basically used to get the distance of the two objects where the angles and an objects distance from the ground are known to us. Its an angle that is formed with the horizontal line if the line of sight is downward from the horizontal line.

If the object observed by the observer is below the level of the observer, then the angle formed between the horizontal line and the observers line of sight is called the angle of depression. In the below figure, is the angle of depression.

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## How To Find The Angle Of Depression

In Figure 11-3, Mr. Toni, from the top of the building, is seeing his friend lying on the ground to take some rest. The height of the building is $70$ m. His friend is $70$ m from the building. Let us determine the angle of depression between Tonis line of sight to his friend and the horizontal line drawn from Tonis eyes.

In this example, the angle $\theta$;represents the angle of depression between Mr. Tonis line of sight to his friend and the horizontal line. Note that the angle of depression is outside of the triangle and measured from the top ceiling. Also, the horizontal line is parallel to the ground surface.

Similarly, note that $CBA$ is an angle of elevation as it is measured from the ground, the angle with what Tonis friend will be looking at him from the ground surface .

Now, we have:

• Two parallel lines $CD$ and $AB$
• A line of sight $BC$ is the transversal

We must recall the geometry that when two parallel lines $AB$ and $CD$, are cut by a transversal line $BC$, we get the alternate interior angles which are angle $\theta$ and $CBA$ in our case. We know that alternate interior angles are congruent. Thus,

Angle of depression $\theta =$ Angle of elevation $CBA$

Now utilizing this fact, we need to label $CBA$ as $\theta$ inside the triangle, as shown in Figure 12-4 below.

Now from the perspective of $mB = \theta$, we observe that:

Opposite side $AC = 70$ m

Adjacent side $AB = 70$ m

Using the formula of the tangent function

$} }}}$

$}}$

$\tan \theta = 1$

## Angles Of Elevation And Depression 5 Powerful Examples

So weve become experts in finding and sketching angles and reference triangles, but what are they good for?

How do people use Reference Angles and Triangles in real life?

Right Triangles are used in all types of architecture. You see them every time you climb stairs, look at a roof, or cross over a bridge.

You see triangles in kites, skateboard ramps, street signs, airplanes, sailboats, and even mountains.

And, havent you ever eaten a sandwich that was first cut into two triangles? Mmmmm, food!

Gosh, triangles are everywhere!

Did you know that you create triangles and use math with your eyes every single day!

Every time you look up at something in the sky, you are creating something called the Angle of Elevation with your eyes. Moreover, when you look down at something on the ground, you are creating an angle called Angle of Depression.

Neat!

Angle of Elevation

Well, trigonometric functions are used to calculate distances by finding an angle determined by a horizontal and a line of sight .

When we elevate our eyes to look up at the top of a building or see a bird in the sky we create an angle with the ground that we can then use to calculate the height or even the distance away from whatever it is we are looking toward. You never know when you might need to calculate the height of a saguaro cactus while driving through Arizona, as Purple Math beautifully illustrates.

Finding lengths and angles of a Right Triangle

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## Find The Angle Of Depression

The angle framed by the line of sight and the even plane for an object below the horizontal is known as the angle of depression. Estimate the same from the known values of opposite and the adjacent sides of the object using this angle of depression calculator.

#### Formula:

Where,

Tangent is the main ratio that is used to determine the angle of depression. It may be found by using this equation tan y is equal to opposite divided by the adjacent side. The opposite side, in this case, is usually the height of the observer or height in terms of location. At the angle of depression, the observer’s line of sight would be above the horizontal. In simple, If you are viewing at an object below the horizon then the angle between the horizontal and your line of sight is the angle of depression. Just enter the measurements of opposite side and adjacent side in this angle of depression calculator to get the result.

## Angle Of Elevation And Depression

Univ. of WisconsinJ.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

In real world situations we often discuss the angle of elevation and depression. The angle of elevation and depression is used often in word problems, especially those involving a persons line of sight as they look up at an object. These angles can be used to solve problems involving trigonometric functions such as sine, cosine, and tangent, and the inverse trigonometric functions.

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