# What are some Real Life Applications of Trigonometry?

Trigonometry is the branch of mathematics that deals with the relationship of sides with angles in a triangle. With trigonometry finding out the heights of big mountains or towers is possible, also in astronomy, it is used to find the distance between stars or planets and is widely used in physics, architecture, and GPS navigation systems. Trigonometry is based on the principle that **“If two triangles have the same set of angles then their sides are in the same ratio”**. Side length can be different but side ratios are the same.

### Trigonometric Functions

- sin A = Perpendicular / Hypotenuse
- cos A = Base / Hypotenuse
- tan A = Perpendicular / Base
- cot A = Base / Perpendicular
- sec A = Hypotenuse / Base
- cosec A = Hypotenuse / Perpendicular.

### Applications of Trigonometry

Trigonometry has huge applications in calculating distances, finding paths in motion, and studying waves, some of them are discussed below:

**Trigonometry is used in measuring the heights of mountains, towers, or buildings:**

Heights of mountains tall towers can easily be found using trigonometry, If you want to find the height of a tower measuring the horizontal distance from the base of the tower and find the angle of elevation to the top of the tower using a sextant, then you can easily find the height of the tower, mountain or any other thing.

**Trigonometry used in construction sites:**

In construction site trigonometry is used to calculate the following:

- Measuring grounds lots, and fields,
- Measuring ground surfaces,
- Making building perpendicular and parallel,
- Roof inclination and roof slopes,
- Installing ceramic tiles and stones,
- The height and width of the building.
- light angles and sun shading.

**Trigonometry used by flight engineers:**

Trigonometry is used to decide the path of an airplane, from landing to take off in calculation of speed, direction, and slope trigonometry is used. During landing and take off what angle and what speed is perfect even when the wind is blowing is calculated using trigonometry.

**Trigonometry in finding paths of moving objects:**

Trigonometry is used in radar systems to calculate the direction and speed of moving objects. Trigonometry also plays an important role in projectile motion, finding paths of bullets, finding the path of a rocket fired, or stone thrown.

**Trigonometry in physics and mathematics:**

In physics and mathematics trigonometry is used in vector algebra, finding components of a vector, cross product, calculus, waves, and oscillations, circular motions, optics.

**Trigonometry in the satellite navigation system:**

A satellite navigation system provides you your location on the map with help of 24 satellites in earth orbit, in the calculation involved trigonometry is used especially the **law of cosine** is used to make calculations simple.

Some more uses:

Trigonometry is also used in Astronomy, Navigation system, Surveying, Architecture, CT scans, and ultrasounds, Number theory, oceanography, computer graphics, and in-game development.

**Some important trigonometric formulas:**

The following identities are obtained using Pythagoras theorem and are true for all values of Angle A.

**sin**^{2}(A) + cos^{2}(A) = 1**tan**^{2}(A) + 1 = sec^{2}(A)**cot**^{2}(A) + 1 = cosec^{2}(A)

### Sample Problems

**Question 1. If the sin of angle A is 0.3, find cos of angle A?**

**Solution:**

Given sin(A) = 0.3

from trigonometric identities we have sin

^{2}(A) + cos^{2}(A) = 1(0.3)

^{2 }+ cos^{2}(A) = 1cos

^{2}(A) = 1 – 0.09cos

^{2}(A) = 0.01cos(A) = 0.10

**Question 2. If the cos of angle A is 0.5, find tan of angle A?**

**Solution:**

Given cos(A) = 0.5

sec(A) = 2

from trigonometric identities we have tan

^{2}(A) + 1 = sec^{2}(A)tan(A)

^{2 }+ 1 = (2)^{2}tan(A)

^{2 }= 3tan(A) = √3

**Question 3. If the cot of angle A is 3, find cosec of angle A?**

**Solution:**

Given cot(A) = 3

from trigonometric identities we have cot

^{2}(A) + 1 = cosec^{2}(A)3

^{2 }+ 1 = cosec^{2}(A)cosec

^{2 }(A) = 4cosec(A) = 2

**Question 4. If the cos of angle A is 0.2, find tan of angle A?**

**Solution:**

Given cos(A) = 0.2

sec(A) = 5

from trigonometric identities we have tan

^{2}(A) + 1 = sec^{2}(A)tan(A)

^{2}+ 1 = (5)^{2}tan(A)

^{2}= 24tan(A) = 2√6

**Question 5. Proof that sin ^{2 }(A) + tan^{2(}A) = sec^{2}(A) – cos^{2}(A).**

**Solution:**

Given: sin

^{2 }(A) + tan^{2}(A) = sec^{2}(A) – cos^{2(}A).Rearranging

sin2(A) + cos

^{2}(A) = sec^{2}(A) – tan^{2}(A)from identities we have sin

^{2}(A) + cos^{2}(A) = 1and sec^{2}(A) – tan^{2}(A) = 11 = sec

^{2}(A) – tan^{2}(A)LHS = RHS

Hence proved!

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