Kiss, Marry, Kill: Sin, Cos, Tan

By Hazel — ON Apr 17, 2025

Mathematics often presents us with fascinating relationships, and the trigonometric functions of sine (sin), cosine (cos), and tangent (tan) are some of the most essential components of trigonometry. These three functions are fundamental to understanding the properties of triangles, waveforms, and circular motion, among many other applications. The interplay between sin, cos, and tan has intrigued students and professionals for centuries. In this article, we will explore the Kiss, Marry, Kill analogy applied to these three functions and examine why each of them is unique, indispensable, and indispensable in its own way.

The Basics of Trigonometric Functions

Before we dive into the analogy, it’s essential to understand the core definitions of these functions. Trigonometric functions describe relationships between the angles and sides of a right triangle.

Sine (sin): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. In a unit circle, sin(θ) gives the y-coordinate of a point on the circle corresponding to an angle θ.
Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, cos(θ) gives the x-coordinate of a point on the circle corresponding to an angle θ.
Tangent (tan): The tangent of an angle is the ratio of the sine to the cosine of that angle. Mathematically, tan(θ) = sin(θ)/cos(θ). This function represents the slope of the line formed by the angle in a right triangle.

Now that we have the basics, let’s analyze these functions and apply the Kiss, Marry, Kill framework to understand their respective roles.

Kiss: Sine – The Most Romantic of the Functions

If we were to “kiss” one of these trigonometric functions, it would undoubtedly be sine. Why? Because sin is arguably the most loved function in the world of trigonometry.

Why Kiss Sine?
The sine function is the most visually appealing of the three. In the unit circle, the graph of sin(θ) is smooth and periodic, oscillating between -1 and 1 as the angle increases. This smooth periodicity is often compared to the natural flow of a wave, making it a popular choice in various fields such as physics, signal processing, and engineering.
Applications of Sine:
From sound waves to electromagnetic waves, the sine function is often used to model periodic phenomena. Its simple yet effective representation of oscillations makes it a key player in trigonometry. It’s used to calculate everything from acoustic frequencies to radio waves, and its importance can’t be overstated.
In the Unit Circle:
In terms of the unit circle, sin(θ) has a straightforward and understandable role: it directly measures the vertical position of a point. Because of this, it is often the first function introduced to students, making it the easiest one to “fall in love” with.

Marry: Cosine – The Constant Companion

If there’s one function that you would want to marry, it would be cosine. While sin is the first love of many, cosine is the steady and reliable partner who always has your back.

Why Marry Cosine?
The cosine function works seamlessly with sine to provide a complete picture of trigonometric relationships. While sin oscillates with a wave-like motion, cos follows right along with it, shifting the graph by 90 degrees. The cosine function’s symmetrical relationship with sine makes it indispensable. If you ever need to find one, you can always rely on the other, making them the perfect duo.
Applications of Cosine:
The cosine function is used in a wide variety of applications, including engineering, physics, and architecture. In addition to its connection with sin, cosine is essential in analyzing motion, particularly in harmonic oscillations. It also plays a pivotal role in the analysis of electrical circuits, particularly in alternating current (AC) analysis, making it a crucial function for electrical engineers.
In the Unit Circle:
On the unit circle, cos(θ) gives the horizontal position of a point. The symmetry of cosine with respect to the x-axis means that it behaves predictably and smoothly, making it the most “stable” of the three functions.

Kill: Tangent – The Wild Card

And then, there is tangent – the function that can be both fascinating and frustrating. If we had to “kill” one of these functions, it would be tan. However, it’s important to note that this is more of a “tough love” scenario. While tan is incredibly useful in certain situations, it does come with its challenges.

Why Kill Tangent?
The tangent function has one major drawback: its asymptotes. Unlike sine and cosine, which smoothly oscillate, tan(θ) can explode to infinity at certain points. This makes it much more unpredictable and harder to handle in certain calculations. The function is undefined wherever cos(θ) = 0, which occurs at θ = π/2 + nπ for any integer n.
Applications of Tangent:
Despite its unpredictability, tan(θ) is still widely used. It’s especially useful when analyzing the slope of a line, since tan(θ) is the slope of the line that makes an angle θ with the x-axis. In physics, tan(θ) is frequently used to model angles of elevation and depression in projectile motion.
In the Unit Circle:
tan(θ) doesn’t represent a specific coordinate in the unit circle like sin or cos do. Instead, it represents the ratio of the y-coordinate (sine) to the x-coordinate (cosine). This lack of direct representation on the unit circle is why tan(θ) can sometimes feel like the rebellious function that doesn’t quite fit into the same neat, well-behaved category as sin and cos.

The Importance of Trigonometric Functions

Although we’ve “killed” tan(θ) in the context of this analogy, it’s important to emphasize that all three functions – sin, cos, and tan – are crucial to understanding the world of trigonometry and mathematics. Each function serves its own unique purpose, and without any of them, the mathematical universe would not be as complete.

Sine and cosine are fundamental to understanding the circular motion of objects and the periodic nature of waves.
Tangent provides essential insight into the slopes of lines and angles, making it a valuable tool for architects, engineers, and physicists.

Conclusion

In conclusion, the functions sine, cosine, and tangent each play an essential role in the world of mathematics. While we may have assigned them roles in the Kiss, Marry, Kill analogy, it’s clear that all three are indispensable. Sine may be the most romantic and appealing, cosine the steady companion, and tangent the wild card that adds a touch of unpredictability. Together, they form the cornerstone of trigonometry and are essential to countless real-world applications.