Trigonometry can be simply defined as calculations with triangles involved with the study of lengths, heights, and angles. Trigonometry and its functions have an enormous number of uses in our daily life. For instance, it is used in geography to measure the distance between landmarks, in astronomy to measure the distance of nearby stars and also in the satellite navigation system.

Before going into the details of trigonometry applications, let’s start with knowing the trigonometry and its functions.

## What is Trigonometry?

Trigonometry is a study of the relationship of angles, lengths, and heights. It has emerged in the third century BC involving applications from astronomy to geometric studies. Now it has spread its application into wider fields like engineering, physics, surveyors, architects, astronauts and even in the investigation of a crime scene.

Apart from astronomy and geography, trigonometry is applicable in various fields like satellite navigation, developing computer music, chemistry number theory, medical imaging, electronics, electrical engineering, civil engineering, architecture, mechanical engineering, oceanography, seismology, phonetics, image compression and game development.

## Applications of Trigonometry:

It may not have direct applications in solving practical issues but used in the various field. For example, trigonometry is used in developing computer music: as you are familiar that sound travels in the form of waves and this wave pattern through a sine or cosine function for developing computer music. Here are few applications where trigonometry and its functions are applicable.

### Trigonometry to Measure Height of a building or a mountain:

Trigonometry is used to in measuring the height of a building or a mountain. The distance of a building from the viewpoint and the elevation angle can easily determine the height of a building using the trigonometric functions.

**Example: The distance from where the building is observed is \( \small 90 ft \) from its base and the angle of elevation to the top of the building is \( \small 35^{\circ } \). Now find the height of the building.**

**Solution:**

Given:

- Distance from the building is \( \small 90 feet \) from the building.
- The angle of elevation from to the top of the building is \( \small 35^{\circ } \).

To solve and find the height of the tower by recalling the trigonometric formulas. Here, the angle and the adjacent side length are provided. So, using the formula of \( \small \tan \).

\( \tan 35^{\circ } = \frac{Opposite \; Side}{Adjacent \; Side} \)

\( \tan 35^{\circ } = \frac{h}{90} \)

\( \small h = 90 \times \tan 35^{\circ} \)

\( \small h = 90 \times 0.4738 \)

\( \small h = 42.64 \; feet \)

Thus, the height of the building is \( \small 42.64 feet \).

### Trigonometry in Aviation:

The aviation technology has been evolved in many up-gradations in the last few years. It has taken in account that the speed, direction, and distance as well as have to consider the speed and direction of the wind. The wind plays a vital role in when and how a flight will travel. This equation cab is solved by using trigonometry.

For example, if an airplane is traveling at 250 miles per hour, \( \small 55^{\circ } \) of a north of east and the wind blowing due to south at 19 miles per hour. This calculation will be solved using the trigonometry and find the third side of the triangle that will lead the aircraft in the right direction.

### Trigonometry in Criminology:

Trigonometry is even used in the investigation of a crime scene. The functions of trigonometry are helpful to calculate a trajectory of a projectile and to estimate the causes of a collision in a car accident. Further, it is used to identify how an object falls or in what angle the gun is shot.

### Trigonometry in Marine Biology:

Trigonometry is often used by marine biologists for measurements to figure out the depth of sunlight that affects algae to photosynthesis. Using the trigonometric function and mathematical models, marine biologists estimate the size of larger animals like whales and also understand their behaviors.

### Trigonometry in Navigation:

Trigonometry is used in navigating directions; it estimates in what direction to place the compass to get a straight direction. With the help of a compass and trigonometric functions in navigation, it will help to pinpoint a location and also to find distance as well to see the horizon.

## Other Uses of Trigonometry:

- The calculus is based on trigonometry and algebra.
- The fundamental trigonometric functions like sine and cosine are used to describe the sound and light waves.
- Trigonometry is used in oceanography to calculate heights of waves and tides in oceans.
- It used in the creation of maps
- It is used in satellite systems.

Thus here we have discussed Trigonometry and its importance as every student of math is expected to know about the application of this branch of mathematics in daily life. Solve sample questions with answers and cross check your answers with the NCERT Solutions on some applications of Trigonometry.

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