# Trigonometric Applications

What is Trigonometry?

The trigonometry word is derived from the Greek word ‘trigonon’ which means triangle and ‘metron’ which means measure. It is the 16th century Latin derivative. The concept of trigonometry was given by Greek Mathematician Hipparchus. According to Victor Katz in “A History of Mathematics (3rd edition)”, trigonometry developed primarily from the needs of Greek and Indian Astronomers.

Trigonometry is the most important concept in Mathematics. It plays an important role in almost all the fields whether its Aviation, Physics, Criminology, Military, etc. Trigonometry is used for finding the angles or the sides of the triangle.

Right- Angled Triangle:

Here in the figure, a right- angled triangle is shown having hypotenuse (the longest side), base (adjacent), height (opposite), and angle.

This triangle is of great importance because if anyone wants to find the direct distance

and angle then that can be easily found using this as shown below.

The basic functions of trigonometry are sine, cosine, and tangent. The other three cosecant, secant, and cotangent are the reciprocal of sine, cosine, and tangent respectively.

Trigonometric ratios are:

Trigonometric Angles:

In trigonometry, there are five angles. Other angles can also be found but these values are basic. These angles are 00, 300, 450, 600, 900 . Below table is given.

 Angles $0^{\circ }$ $30^{\circ }$ $45^{\circ }$ $60^{\circ }$ $90^{\circ }$ sin $\theta$ 0 $\frac{1}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{\sqrt{3}}{2}$ 1 cos $\theta$ 1 $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{2}}$ $\frac{1}{2}$ 0 tan $\theta$ 0 $\frac{1}{\sqrt{3}}$ 1 $\sqrt{3}$ $\infty$ cosec $\theta$ $\infty$ 2 $\sqrt{2}$ $2 \sqrt{3}$ 1 sec $\theta$ 1 $2 \sqrt{3}$ $\sqrt{2}$ 2 $\infty$ cot $\theta$ $\infty$ $\sqrt{3}$ 1 $\frac{1}{\sqrt{3}}$ 0

Unit Circle:

The concept of unit circle is used to find the angle. Since the circle is at origin and have unity radius it is easy to directly find the sine, cosine and tangent.

Trigonometric Formula:

Trigonometric identities and formulae are based on right- angled triangle:

1. Pythagorean Formula

1. Sum and Difference Identities:

1. If A, B, and C are angles and a, b, and c are the sides of a triangle then,

Trigonometry Applications in Daily Life:

Trigonometry has a wide range of applications in our everyday life. Trigonometry is used in calculus, astronomy, aviation, etc.It has vital uses in the military and also in Marine Biology. The fields where trigonometry is used are

• Oceanography

• Meteorology

• Seismology

• Acoustics

• Electronics, etc.

It is also helpful to measure the height of the mountain, pillar, etc.

The use of Trigonometry to Measure the Height of a Mountain or a Building:

From the angle of elevation, the height (z) can easily be measured.

Applying the trigonometric ratios for finding z in the figure above,

If the angle is known, then,

This is how the height can be measured.

Basically, the height of the mountain or building can be easily measured using the trigonometric ratios. Consider the base of the mountain or building as adjacent and the height as opposite, then the third longest part of the triangle will be hypotenuse as shown above in the figure. Triangle should be right- angled triangle. Hence, this is the real-life application of trigonometry.

Trigonometry in Aviation:

Incase of aviation, wind plays a most important role. In aviation the angle of depression and the angle of elevation is used depending on the case. Angle of depression is when the line of sight to the object is below the horizontal line and the angle of elevation is when the line of sight to the object is above the horizontal line.

The wind of flight direction is considered as the two perpendicular sides of a right- angled triangle in which the speed of the flight and the speed of wind are measured in their direction.

Explanation: From the above figure, the distance between the plane and the final point is one of the perpendicular distance. The distance of the plane from the ground is another perpendicular distance. Slant distance is the hypotenuse.

For this, trigonometric ratios are used for finding the distance or angle of aviation.

Trigonometry in Criminology:

In criminology, trigonometry can be used to find the shots of bullet or how tall the shooter was. This helps the police in the investigation. For this also trigonometric ratios are used i.e., sine, cosine, tangent. This is also the real-life application of trigonometry.

From the figure, it can be clearly seen that the triangulation is done through which the distance and angle can be found.

Trigonometry in Marine Biology:

Marine Biology is the study of marine life. In marine biology, the biologists study about marine life like about plant, animals and other organisms that live there.

In marine biology, for predicting how far and at what angle the organism or plant is from the observer, trigonometry concepts are used.

• Here, the water level is the base of the right-angled triangle and

• the depth of the object is the height

• slant distance is the line of sight

• Angle could be angle of depression or angle of elevation, it depends on the observer who is observing.

Hence, trigonometry ratio and formula are used for finding the distance, angle.

Hence the exact location of the underwater living or non- living things can be measured easily with the help of trigonometry ratios sine, cosine and tangent. Hence, this is again a real life application of trigonometry.

In navigation, controlling and monitoring of moving vehicles are studied. In this, the movement of the plane, ships, submarines, etc is studied. Navigation is of many types- marine navigation, aeronautical navigation, space navigation, and land navigation. Navigation is the study of moving objects and the angle or distance is measured using the concept of trigonometry at that point where we have to measure.

To measure the exact location of the moving body, compass, pinpoint poles, trigonometric ratios and formula are used.

Other Uses of Trigonometry:

Trigonometry is used everywhere.

• Trigonometry can be used in video games. Mario the most famous video game is based on trigonometric functions. Its jump, trajectory all are measured using trigonometry.

• Trigonometry is also used in sound waves and light waves.

• In construction, trigonometry is widely used for inclination of the roof. Trigonometry is used to define the relationship between the sides of the roof and the inclination of the roof.

• In the creation of any geographical map, trigonometry is used.

• In satellite, trigonometry is used. Where the satellite is and at what angle this all can be easily measured using the trigonometry. Exact location or position of the satellite can be easily measured using the trigonometric ratios.