February 15, 2024

Kathlyn Phan

11th Grade

Fountain Valley High School

In the complex world of trigonometry, there are three key functions: sine, cosine, and tangent. These functions help us solve problems and better our understanding of right-angled triangles. But how were they created? These mathematical tools stemmed from ancient mathematicians and astronomers deriving them from side length ratios in right-angled triangles. By understanding their history, we can see how sine, cosine, and tangent evolved from ancient problem-solving techniques to the essential math tools we use for everyday applications today. 

The word “trigonometry” comes from Latin and means “the study of trigons”, which are triangles. Trigonometric functions are used whenever there is a  problem regarding right triangles. It was created by an ancient Greek mathematician in 190-120 BC named Hipparchus. He is nicknamed “the father of trigonometry” because he was the first to construct a table of values for the creation of trigonometric functions. In his table, he had corresponding values for both arc and chord across numerous types of angles. 

Along with being a mathematician, Hipparchus was also an astronomer. He focused on spherical triangles he observed formed by three celestial stars. Now, there are applications for trigonometric functions across a diverse spectrum of uses. For instance, cartography, engineering, oceanography, astronomy, etc.

Sine is an acute angle in a right-angled triangle whose measure is the ratio of the opposite leg to the hypotenuse. The universal symbol for angle measurements is theta (θ). Sine is considered an odd function. This is because the odd functions’ inverse is f(-x) = -f(x). In the case of sine, it is sin(-θ) = -sin(θ). A common application of sine is the sine rule where you can use the given information of any non-right triangle to calculate the missing angle or side measurements. The sine rule is used to figure out the angle of a tilt in engineering or the distance between stars in astronomy. Each of the three trigonometric functions also has its own reciprocal function. For sine, it is cosecant (csc). Cosecant angles are defined as the ratio of a right triangle’s hypotenuse over its opposite leg, which is the opposite of the sine ratio. However, sine has a periodicity of 2π while cosecant has a periodicity of π. This means that these two functions repeat their values after an interval of 2π or 360 degrees and π or 180 degrees, respectively. 

Cosine is defined as the acute angle in a right-angled triangle whose measure is the ratio of the adjacent leg to the hypotenuse. Unlike sine, cosine is considered an even function. This is because even functions’ inverse is f(-x) = f(x). In the case of cosine, it is cos(-θ) = cos(θ). The cosine rule is used to calculate missing leg lengths or angle measures of non-right triangles that are side-side-side triangles (SSS) or side-angle-side triangles (SAS). These types of triangles cannot be solved using the rule of sine, but it can be solved using the rule of cosine. The reciprocal function of cosine is secant (sec). Secant angle measures are the ratio of the hypotenuse to the adjacent leg of a right triangle. Similar to sine and cosecant, cosine has a periodicity of 2π or 360 degrees while secant has a periodicity of π or 180 degrees. When graphed, their values will start to repeat after these intervals. 

Tangent is the acute angle in a right-angled triangle that is the ratio of the opposite leg to the adjacent leg. This trigonometric function is different from the previous two functions, sine and cosine, because it doesn’t involve the length of the hypotenuse. Tangent is defined as the ratio of sine to cosine (tan θ = sin θ / cos θ). Unlike sine and cosine, tangent has vertical asymptotes when it is graphed. This means that there are certain x-values that the function cannot have. The reciprocal of tangent is cotangent, which is defined as the ratio of the adjacent leg to the opposite leg of a right triangle. When graphed, both tangent and cotangent have the periodicity of π or 180 degrees, so their values will repeat after this interval. Tangent having the periodicity of π is another characteristic that further sets it apart from sine or cosine because they have the periodicity of 2π. Tangent is commonly used to calculate the slope of a line connected from the origin to the point representing the altitude of a right triangle.

In conclusion, sine, cosine, and tangent serve as key players in trigonometry to serve as vital tools for solving any type of triangle, right or non-right, along with their reciprocal functions. From ancient problem-solving techniques to modern-day applications, trigonometric functions are utilized in various aspects of engineering, astronomy, oceanography, etc. These mathematical tools reveal the importance of understanding the angles of our world.