# How is trigonometry used in game development?

## Introduction

Do you love to play games? Indeed, we all do. So what’s your favourite game? But before that, do you know the ideology behind the development of your favourite game? Well, it is the magic of trigonometry. Yes, you heard it right. Trigonometry gets widely used in game development. You must be pondering how trigonometry is used in-game designing? While studying the concept of trigonometry, we usually face the following problems or questions–

What is trigonometry?

Can we use trigonometry to design a game?

What is the process of developing a game using trigonometry?

What are sin, cos, and tan?

If you have almost similar queries in mind, trust us, then we are sailing the same boat. Many young students face the same problem while comprehending complicated mathematical topics like trigonometry. So let us delve deep into the world of trigonometry and explore the concept thoroughly-

## Meaning

Have you ever thought that games such as GTA 5 and Call of Duty get more realistic and surreal with every new edition? It is due to the magical power of trigonometry principles. Trigonometry is one of the oldest branches of mathematics that explains and assesses the angle and sides of a 90-degree triangle. The idea of trigonometry was propounded by greek mathematician Hipparchus.

Trigonometry evolved from the exigency to calculate the angles in various fields like astronomy, gaming, and artillery, etc. For this reason, it is also called the study of relationships between lengths and angles of triangles. So from next time your car gets crashed in the game, you know the maths behind it. Right?

## Important Trigonometric functions

Here is the list of six trigonometric functions that need to learn and analyzed before apprehending the core of trigonometry-

 Function Acronym Relationship to the side of the right triangle Sine Function sin Opposite side/ Hypotenuse Tangent Function tan Opposite side / Adjacent side Cosine Function cos Adjacent side / Hypotenuse Cosecant Function cosec Hypotenuse / Opposite side Secant Function sec Hypotenuse / Adjacent side Cotangent Function cot Adjacent side / Opposite side

## Trigonometry Table

Now next comes the trigonometry table, it is helpful in solving the problems related to gaming development and other complex trigonometric equations –

 Angles 0° 30° 45° 60° 90° Sin θ 0 ½ 1/√2 √3/2 1 Cos θ 1 √3/2 1/√2 ½ 0 Tan θ 0 1/√3 1 √3 ∞ Cosec θ ∞ 2 √2 2/√3 1 Sec θ 1 2/√3 √2 2 ∞ Cot θ ∞ √3 1 1/√3 0

## Trigonometry in game development

In most of the games, trigonometric functions get implemented to obtain desirable results. In order to calculate the angle of the triangle with 900 degrees corner, various trigonometric functions are usually called for action. Do you know the online gaming industry in India is expected to grow at a CAGR of 22% by FY23?

For understanding the process of developing a game using trigonometry, a few prime concepts that you need to get well-versed. They are –

• Tangent – The ratio of the opposite side to the adjacent side is called Tan or Tangent. Let us try to understand it with the help of an example –

Suppose angle α needs to get calculated in the triangle. X is 10, and y is 6. We will get the following equation –

Tan( α ) = y / x

Since we want to get the value of α, we will do Tan-1 on both sides of the triangle.

α =Tan-1 ( y / x )

Now we can calculate by inserting the values and will get the following equation –

α =Tan-1 ( y / x )

α = Tan-1 ( 6 / 10 )

α = Tan-1 ( .6 )

α = 31⁰

• Cos- The ratio of the adjacent side of a right triangle to the hypotenuse termed as Cos.

When only the values of x and z will be given, we use the cos function to find the value of α . This is similar to a tangent function. Here, x = 10 and z = 12

Cos( α ) = x / z

Again here also Cos-1 on both sides of the equal sign will get used.

α =Cos-1 ( x / z )

Now putting the values of x and z, following equation will popped up –

α =Cos-1 ( 10 /12 )

α =Cos-1 ( .8 )

α =34⁰

• Sine- The ratio of the opposite side of a right triangle to the hypotenuse called Sine. Now, x = 6 and z = 12. Since the function is almost similar to cos and tan, input values are different here. The equation follows-

Sin( α ) = y / z

Similarly, we employed both sides of the equal sign and the function Sin-1

α =Sin-1 ( y / z )

Lastly, we will put the values like we used to do in the previous examples. After that, we will compute. The following equation will get appeared –

α =Sin-1 ( 6 /12 )

α =Sin-1 ( .5 )

α =30⁰

At first, these functions may appear confusing and intimidating, but with practice, one can easily excel in these functions. These elements get adopted in game development. Now the question arises how these concepts get implemented while fabricating games?

Suppose you have two sides of the triangle, we need the angle in it.

For Example :- Cos(A)=z/y

With z and y, we can say that the angle, A, can be found when the cosine equals z/y. But for gaming, ATan is mostly used.

A = ACos(z/y)

## Mouse Triangle

360-degree shooting can be seen in most of the game that is due to trig. When we want to shoot an object in the direction of X and Y of the triangle and your mouse. Xmouse-X(“Object”) and y = Ymouse-Y(“Object”) makes a triangle with Z as hypotenuse.

Usually, to calculate the direction of players, tan gets used.

Tan( Dir ) = Opposite / Adjacent

It gives us the equation:

Tan( Dir ) = xSpeed /ySpeed

Now by using the inverse function of tan, we get the equation that returns the player’s direction

## Calculating player xSpeed and ySpeed

In the previous example, we learned how trigonometric function – tan could be used to identify the direction of the player with the xSpeed and ySpeed as given values. Now we will see how trigonometry gets utilized to find out the speed of the player in the given direction.

By speed here, we refer to the longest side of the triangle. First of all, the sides of the triangle need to be defined, and speed can get denoted by Hypotenuse. Following function gets surfaced-

Sin( α ) = Opposite / Hypotenuse

Cos(α ) = Adjacent / Hypotenuse

Tan(α ) = Opposite / Adjacent

Ultimately, this equation gives us the speed of the players in the game.

Cos( α ) = Adjacent / Hypotenuse

Cos( Dir ) = ySpeed / Speed ySpeed = Cos( Dir ) * Speed

From the above-detailed explanation, we can figure out that every aspect of the game gets developed through trigonometry. With the help of trigonometric functions, one can develop amazing games with realistic effects and exemplary interfaces.

## Importance of trigonometry in game development

• Trigonometry holds an essential role in game development, and it helps the game to function smoothly and effectively.
• In order to move the objects in the game, the trig function gets employed.
• It can also be used in defining sets, characters, and background.
• While developing a game, the positions and speed of the objects should be engraved as X and Y. But finding the value of positions and speed, triangles are used, and trigonometry becomes necessary to identify the value of x and y so that it can be put in code for the game to function.

Conclusion

So we can conclude that trigonometry has a cardinal role to play in game development. More specifically, sin, cos, and tan get widely used for gaming. Since now you have better cognizance about the subject matter, you will feel more confident to answer the questions related to trigonometry. With the help of Trigonometry, you can solve complex design problems and come up with innovative and creative solutions. Trigonometry has a wide array of uses in various disciplines like physics, 3D modelling, architecture, dynamic study, and kinematics. Trigonometry is one of the salient chapters for your mathematics exam, and by checking some cool Math exam tips from Cuemath, you can fetch more marks in your next math exam. There are a plethora of benefits of Trigonometry which is not just limited to the gaming world, once you explore different aspects of trigonometry, you will find that the aegis of trigonometry is so wide and deep.

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