Trigonometry is an essential part of mathematics. It is the study of angles and triangles. Trigonometry basically deals with sides and angles of right triangles. It learns about the relation between the angles and the ratio of sides of right angled triangle. There is an important topic in trigonometry known as heights and distances. It is an applied trigonometry and is used in various fields.

The ** angle of elevation** is a widely used concept related to heights and distances. An angle of elevation is defined as an angle between the horizontal plane and oblique line from the observer's eye to some object above his eye. Eventually, this angle is formed above the surface. As the name itself suggests, the angle of elevation is so formed that it is above the observer's eye. Solving angle of elevation is used in finding distances, heights of buildings, towers etc with the help of trigonometric ratios.

The angle of elevation is similar to the angle of depression (the angle of depression is the angle between the horizontal plane and oblique line joining the observer's eye to some object beneath the line of his eye).Let us learn more about angle of elevation and its applications in real life problems.

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Angle of elevation is an angle that is formed with the horizontal. If the line of sight is upward from the horizontal, the angle is an angle of elevation. In other words, the angle above horizontal that an observer must look to see an object that is higher than the observer is called angle of elevation. In this figure, $\theta$ is the angle of elevation.

The formula for finding the angle of elevation is depending on knowing the information for opposite, hypotenuse, and adjacent to the right angle. If the distance from the object and height of object is given, then the formula for the angle of elevation is given by:

Tangent of angle of elevation = $\frac{\text{Height of object}}{\text{Distance from object}}$

The formula for angle of elevation is also called the altitude angle. It measures the angle of the sun in relation to a right angle. The elevation is the measurement of the angle from the horizon to the line drawn from the sun to a right angle.

### Solved Examples

**Question 1: **From the figure, find the value of $\theta$.

** Solution: **
**Question 2: **In right triangle ABC, if $\angle$ C = 90^{o}, $\angle$ A = 45^{o} and AC = 21. Find the value of BC.

** Solution: **
### Solved Examples

**Question 1: **

** Solution: **

**Question 2: **Building A and building B are on horizontal ground. The angle of elevation of the top of building A from that of the lower building B is 15^{o}. Building A is 100 m in height and B is 80 m in height. Find the distance between both the buildings.

** Solution: **

The formula for finding the angle of elevation is depending on knowing the information for opposite, hypotenuse, and adjacent to the right angle. If the distance from the object and height of object is given, then the formula for the angle of elevation is given by:

Tangent of angle of elevation = $\frac{\text{Height of object}}{\text{Distance from object}}$

The formula for angle of elevation is also called the altitude angle. It measures the angle of the sun in relation to a right angle. The elevation is the measurement of the angle from the horizon to the line drawn from the sun to a right angle.

Mainly two terms are related to the angle of elevation:

- Angle
- Horizontal Line

**Angles**

When two rays or two line segments join at a common endpoint called the vertex. And two straight lines meet at a point is said to form an angle. Angle play an important role in the trade and definitions are:

- Two straight lines meeting at a point form an angle.
- The angle is a gap in between two line which connect on one side.
- The space measured in degrees.

**Horizontal Line**

A straight line on the coordinate flat surface where all points on the line have the same y-coordinate. The angle and Horizontal line both combine to form terms in angle of elevation.

In triangle ABC, AC = 335 ft and BC = 249.

Find $\angle$ A = $\theta$

Now $\tan \theta$ = $\frac{249}{335}$

= .74(approx)

$\theta$ = arc tan(.74)

= 36

$\theta$ = 36.

Find $\angle$ A = $\theta$

Now $\tan \theta$ = $\frac{249}{335}$

= .74(approx)

$\theta$ = arc tan(.74)

= 36

$\theta$ = 36.

Given $\angle$ C = 90^{o}, $\angle$ A = 45^{o} and AC = 21.

Now, tan A = $\frac{BC}{AC}$

tan 45^{o} = $\frac{BC}{21}$

1 = $\frac{BC}{21}$

[tan 45^{o} = 1]

BC = 1 * 21

BC = 21.

Now, tan A = $\frac{BC}{AC}$

tan 45

1 = $\frac{BC}{21}$

[tan 45

BC = 1 * 21

BC = 21.

Given below are some of the word problems on ** angle of elevation**.

A girl is sitting in the shade under a tree that is 90 ft from the base of a tower. The angle of elevation from the girl to the top of the tower is 35 degrees. Find the height of the windmill.

** ** The girl is 90 feet from the tower

The angle of elevation from the girl to the tower is 35^{0}

**Step 1:**Here, we want to solve and find the height of the tower.

Recall the trigonometric formulas.

The angle and the adjacent side length is given.

So, use the formula of tan is

tan 35^{0} = $\frac{\text{Opposite side}}{\text{Adjacent side}}$

tan35^{0} = $\frac{h}{90}$

h = 90 * tan 35^{0}

h = 90 * 0.4738

h = 42.64 feet

Thus, the height of the tower is 42.64 feet.

Below diagram represent the given situation:

Let the distance between both the buildings = x m

Since AE = 100 m

then DE = 100 - 80 = 20 m

and AB = DC = x m

In triangle DCE, Using trigonometric ratio we have,

tan 15^{o} = $\frac{DE}{DC}$

tan 15^{o} = $\frac{20}{x}$

0.26 = $\frac{20}{x}$

x = $\frac{20}{0.26}$

= 76 m (approx)

The distance between two buildings is 76 m.

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