Sine, Cosine, and the Unit Circle: A Story of Circles and Waves

Estimated reading time: 6 minutes

When students learn trigonometry, they often ask what sine and cosine functions are. They also wonder why the graphs of these functions always show up when people talk about waves, and why they are important. Initially, trigonometry might appear like a complicated web of identities, angles, and formulae, but it all depends on three basic ideas: Sine, Cosine waves, and Unit Circle. These basic ideas stand as the fundamental pillars of trigonometric foundation.

What are Sine, Cosine, and Unit Circle?

For the right angle triangle, the basic definition of these functions is as follows:

Sine:- The ratio of the length of the opposite side to the hypotenuse side is the Sine of an angle.

Cosine:- The ratio of the length of the adjacent side to the hypotenuse side is the  Cosine of an angle.

Mathematicians wanted a more general definition that applied to all angles, not only those in right triangles, but both definitions function well within triangles. Which brings us to the unit circle, which is a circle with a radius of exactly 1 unit. 

Unit Circle: A circle centered at the origin (0,0) of a coordinate plane with a radius of one constitutes the unit circle. 

The x-coordinate, the horizontal position, creates a cosine function as the path is traced around the circle. Likewise, the y-coordinate, a measure of vertical position, produces a sine function as the tracer follows its course around the circumference. It is known that using sine and cosine waves, it is possible to represent any (x,y) point on the unit circle with x = cos(θ) and y = sin(θ).

Since sine and cosine are so easy for the unit circle, they can be defined for any angle, even the same way as for positive angles (including zero angles) and angles greater than 90 degrees.

Circular Coordinates: Sine and Cosine Waves

Let us look at the values of the sine and cosine functions as the angle varies around the unit circle. Let’s think of the angles on unit circles as 0°, 90°,180°, 270°,360°, and so forth. The circular coordinates at those particular angles are:

• At 0°, x = cos(0°) = 1 and y = sin(0°) = 0; the point is therefore (0,1).
• At 90°, x = cos(90°) = 0 and y = sin(90°) = 1; the point is therefore (1,0).
• At 180°, x = cos(180°) = -1 and y = sin(180°) = 0; the point is therefore (-1,0).
• At 270°, x = cos(270°) = 0 and y = sin(270°) = -1; the point is therefore (0,-1).
• At 360°, x = cos(360°) = 1 and y = sin(360°) = 0; the point is therefore (0,1).

Again, at 360°, the point is (0, 1), which is the same as the starting point, so the cycle repeats after that. 

From Circles to Waves:

Plotting the values x and y against the given angles creates wave patterns of sine and cosine. Given that the radius of the unit circle is 1, the minimum and maximum values of the two following functions would therefore be -1 and 1. This shows that the waves also oscillate between -1 and 1.

Periodic phenomena are the occurrences in which a function repeats the values it takes; sine and cosine functions repeat at 360 degrees, or 2π radians, as the values also repeat at that point.

All points (x,y) that lie on the circumference of the unit circle satisfy the equation of the circle: x² + y² =1. By employing the sine and cosine functions, we get the most fundamental trigonometric identity, cos²(θ) + sin²(θ) = 1, which is derived by substituting x = cosθ and y = sinθ.

Real-Life Applications of Sine and Cosine:

Apart from their use in mathematics, sine and cosine have numerous applications in everyday life. These are what architects and engineers use to calculate forces and build stuff. They help to explain patterns of motion and wave patterns in physics. Because sound waves and musical tones share sinusoids, this is also important in computer graphics, for 3D modelling and animation.

Sine, Cosine, and Unit Circle: Conclusion

These relationships to the unit circle and one another are what reveal one of the most beautiful truths in all of mathematics. Simple geometric ideas describe these complex natural phenomena. The circular motion of a point on a circle is the origin of the mathematical terms utilized to describe waves, oscillations, and periodic behavior throughout the universe. 

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FAQs:

Q: What is the range of Sine and Cosine?

Answer: Since sine and cosine waves functions assume their highest and lowest values to be -1 and 1, respectively. Then the range of these functions is [-1,1].

Q: Where are the applications of Sine and Cosine in Real Life?

Answer: The real-life uses of Sine and Cosine waves are in computer animation, aiding astronomers, musicians, and medical (for example MRI imaging).

Q: When plotting sine and cosine graphs, where does the curve come from, and what is the ‘wave’?

Answer: This is because a wave picture naturally arises when you draw the y-coordinate (sine) or the x-coordinate (cosine) of a point moving around the unit circle as a function of the angle.

Q: Why does the unit circle have radius 1?

Answer: Within the definition of the unit circle, the radius is 1. However, it is not just any circle. It serves the purpose of a standard reference for trigonometric functions, sine, cosine, and angles.

References

1. Maknun, C. L., Rosjanuardi, R., & Jupri, A. (2019). From ratios of right triangle to unit circle: an introduction to trigonometric functions. Journal of Physics Conference Series, 1157, 022124. https://doi.org/10.1088/1742-6596/1157/2/022124
2. Martínez-Planell, R., & Delgado, A. C. (2016). The unit circle approach to the construction of the sine and cosine functions and their inverses: An application of APOS theory. The Journal of Mathematical Behavior, 43, 111–133. https://doi.org/10.1016/j.jmathb.2016.06.002