Trigonometry and Its Relationship With Unit Circle

There is nothing like trigonometry which so occupies the middle position of mathematics. If that is true, you might have studied this topic in your school. Can you recall it? No? Let me remind you, the word trigonometry is derived from two Greek words where ‘tri’ means ‘three’, ‘gon’ means ‘sides’ and ‘metron’ meaning ‘measure’. In fact, trigonometry is the study of the relationship between sides and angles of a triangle. Trigonometry was first known and was recorded in Egypt and Babylon. Even today most of the technologically advanced work in physics and mathematics are done through trigonometry.

What is a Unit Circle?

A circle with a radius measuring 1 unit is known as a unit circle. It is generally represented in the cartesian coordinate plane. It is algebraically represented with two variables X and Y. The application of the unit circle is also used in trigonometry to find the values of trigonometric ratios such as sine, cosine, tangent, etc. The unit circle has all the characteristics of a circle, and its equation is also derived from the circle. The general equation of a unit circle is (x – a)(x – a)  + (y – b)(y – b) = r.r. A unit circle is formed with its center at the point (0,0) which is the origin and has a radius of 1 unit.

Unit Circle and Trigonometric Identities

The trigonometric identities of sine, cosecant, and tangent can be further changed to other trigonometric identities such as cotangent, secant, and cosecant using the unit circle. These identities mentioned above are the reciprocal of sine, cosecant, and tangent. Further, we can find the value of tan θ by dividing sin θ with cos θ and we can also find the value of cot theta by dividing cos θ with sin θ.

Finding Trigonometric Functions Using a Unit Circle

To calculate the trigonometric functions of sine, cosine, and tangent, we can use a unit circle. We can apply the Pythagoras formula in a unit circle to find the trigonometric functions. Let us take a right-angled triangle in a unit circle to understand the trigonometric functions. The radius present inside the circle represents the hypotenuse of the right-angled triangle. Here, the values of x and y are the altitude and length of the right triangle. Now, we have a triangle with sides 1, x, y. If we apply this in trigonometry, then we can find the values of trigonometric ratios as follows: sin θ =y/1 and cos θ =x/1. We can also change the values θ in order to give the principal values of these trigonometric ratios.

Applications of Trigonometry

There are various important applications of trigonometry in mathematics especially, but it is also applicable in our day-to-day life. The applications are as follows:

1. Trigonometry is used in developing and composing computer music. With the help of trigonometric functions such as sine and cosine, computer music is produced.
2. The trigonometric identities are widely applicable in measuring the height of a tree, mountain, or building.
3. In most of crime scenes, trigonometry is used in order to calculate the trajectory of a projectile.
4. The whole system of calculus is based on trigonometry and algebra.
5. It is used in the creation of maps, used in satellites, etc.
6. To calculate the height and density of the oceanic waves, trigonometry is used. This process is also known as oceanography.
7. Trigonometry is also used in navigating directions with the help of a compass which shows the direction.

From calculating the height of a mountain to being the base for the subjects like Calculus, trigonometry has immense use. To learn more about trigonometry and unit circle in an interesting and fun way, visit Cuemath.