Trigonometry: An Introduction

Trigonometry is the branch of mathematics regarding specific functions of angles and their implementation in calculations. Mastering trigonometry is crucial for those who aspire to become future mathematicians, find success in high school arithmetic, and to further educate themselves upon the wonders of math. Within introductory trigonometry, it is pertinent to understand the 3 trigonometric ratios.

Firstly, in prefatory trigonometry, it is important to keep in mind that trigonometry only works on right triangles. Later on, we will discover and learn how to utilize trigonometry within triangles that do not have a right angle. To understand the 3 ratios, we must acknowledge the composition of a triangle first. To name the sides of a triangle, we must first base it on an angle. This angle is known as the "reference angle" and can be any angle of your choice, aside from the right angle. Your reference angle can be labeled as anything you would like, but in our case, it is labeled "theta", a common variable used in math (θ). The side of the triangle which is opposite the reference angle is known fittingly as the "opposite" side. The side that is adjacent to the reference angle is known as the "adjacent" side. Lastly, the side which is the longest is known as the "hypotenuse". It is important to note that the adjacent side can never be the hypotenuse, and should always be the side opposite to the hypotenuse. This can be visualized in the image below.

(Math is Fun, 2022)

Next, let us explore the 3 trigonometric ratios. Firstly, we need to establish a reference angle, let's use θ such as in the image above. Next, let us explore the 3 ratios.

The first ratio that we should know is called "sine". Sine is the ratio between the opposite side and the hypotenuse of a triangle, and is often abbreviated as "sin". In an equation, sine would be: sinθ = opp/hyp.

The second ratio that we should know is called "cosine". Cosine is the ratio between the adjacent side and the hypotenuse of a triangle, and is often abbreviated as "cos". In an equation, cosine would be: cosθ = adj/hyp.

The third and last ratio that we should know is called "tangent". Tangent is the ratio between the opposite and adjacent sides of a triangle, and is often abbreviated as "tan". In an equation, the tangent would be: tanθ = opp/adj. Tangent is the only trigonometric ratio that does not involve the hypotenuse.

Another concept to note is what we call "arctangent", "arcsine", and "arccosine". These are simply the reciprocals of their respective ratios and are commonly used when we want to solve for an angle in a triangle. Their respective symbols are tan^-1, sin^-1, and cos^-1. Their equations are θ = tan^-1(opp/adj), θ = sin^-1(opp/hyp), and θ = cos^-1(adj/hyp).

Trigonometry is a crucial field of mathematics and mastering it is pivotal for further education in the subject. After an introductory view of trigonometry's basic concepts, problem-solving and further schooling on the topic will commence in the ensuing blogs.

Maor, E. (2019). Trigonometry. In Encyclopædia Britannica. https://www.britannica.com/science/trigonometry Math is Fun. (2022). Trigonometry [Online Image]. In mathisfun.com. https://www.mathsisfun.com/algebra/trigonometry.html