In the previous guide, you were introduced to the unit circle: a circle with a radius of 1, centered at the origin of a coordinate plane. In this guide, we will explore the patterns present in the unit circle and use those patterns to create and understand the graphs of the sine and cosine functions. 

Patterns in the Unit Circle

Image courtesy of Remind.

Recall that the cosine function gives the x-coordinate of a point on the unit circle as we move counterclockwise around it. An important pattern to identify is that as you go from 0 to 𝛑 radians, the values of the x-coordinate (the cosine values) decrease. This is because the horizontal distance from the origin to our point P on the unit circle decreases as we go through the first two quadrants of the coordinate plane (after 𝛑/2 radians, the cosine values become more negative). After 𝛑 radians, the cosine values start to increase again until we reach 2𝛑 radians. This is because the horizontal distance is increasing as we travel through the third and fourth quadrants. 

It is important to note that the magnitude (absolute value) of the distance is not always increasing, but the values of the x-coordinates (the cosine values) are, since they are going from negative to less negative, and negative to more negative for decreasing cosine values.

Notice how the cosine values range from 1 to -1, and will reach a maximum of 1 at the point (1,0) and reach a minimum of -1 at the point (-1,0). These points correspond to angles of 0 and 𝛑, respectively. At 𝛑/2 radians and 3𝛑/2 radians, the cosine value is 0.

On the other hand, the sine function gives the y-coordinate of a point on the unit circle as we move counterclockwise around it. The important pattern to identify with the sine function is that as you go form 0 to 𝛑/2 radians, the values of the y-coordinate (the sine values) increase. This is because the vertical distance from the origin to our point P on the unit circle increases as we go through the first quadrant of the coordinate plane. After 𝛑/2 radians, the sine values start to decrease until we reach 3𝛑/2 radians. This is because the vertical distance is decreasing as we travel through the second and third quadrants. From 3𝛑/2 radians to 2𝛑 radians, the sine values start to increase again as the vertical distance gets more positive.

The sine values will also range from 1 to -1, and will reach a maximum of 1 at the point (0,1), and reach a minimum of -1 at the point (1,0). These points correspond to angles of 𝛑/2 and 3𝛑/2, respectively. At 0 and 𝛑 radians, the sine value is 0. If you compare these key points to the key points of the cosine function, you'll notice that the points are opposites. When the cosine value is 0, the sine value is at a maximum or minimum, and when the cosine value is at a maximum or minimum, the sine value is 0. It is important that you learn the behavior at the key points of both the sine and cosine functions.

Constructing the Sine Curve

To understand how the patterns we noticed above can be used to construct the sine curve, let's first review what sine is.

Sine is a trigonometric function: a mathematical function that relates the angles of a right triangle to the ratios of its sides. In other words, it is a function that takes an angle as input and returns a value between -1 and 1, which corresponds to the ratio of the side opposite the angle to the hypotenuse. These ratios are represented on the coordinate plane as the y-coordinate of a point on a unit circle.

To construct the sine curve, we can start by plotting the values of the sine function (dependent variable) for different angles on the coordinate plane (independent variable). To do this, we can use the patterns we noticed on the unit circle, where the angle is given in radians, and the corresponding sine value is calculated using the sine formula. The resulting graph will be a smooth curve that oscillates (goes back and forth, like a pendulum) between -1 and 1, since those were the maximum values found on the unit circle for the sine function.

We will plot points starting at 0 radians, where the sine value is 0. This point is plotted at the origin because both the independent and dependent variables are 0, so the point will become (0, 0). Next, we noticed that our maximum sine value occurred at 𝛑/2 radians, so we would plot the point (𝛑/2, 1) on the coordinate plane. We also noticed that our sine value is 0 at 𝛑 radians, making the point (𝛑, 0) an x-intercept. We continue using the points on the unit circle to plot our graph of the sine function, and connect our points using a smooth curve. We can now see that the sine function looks like this:

Image courtesy of Wikimedia.

The sine function can be written as f(θ) = sin(θ) in function notation. It's important to note that the sine function is periodic, meaning it repeats its values at certain intervals. In the case of the unit circle, these intervals are multiples of 2𝛑.

Image courtesy of Study.

Constructing the Cosine Curve

We can use a similar process to construct the cosine curve, but first, let's review what cosine is.

Cosine is a trigonometric function that relates the angles of a right triangle to the ratios of its sides. In other words, it is a function that takes an angle as input and returns a value between -1 and 1, which corresponds to the ratio of the side adjacent to the angle to the hypotenuse. These ratios are represented on the coordinate plane as the x-coordinate of a point on a unit circle.

To construct the cosine curve, we can start by plotting the values of the cosine function (dependent variable) for different angles on the coordinate plane (independent variable). To do this, we can use the patterns we noticed on the unit circle, where the angle is given in radians, and the corresponding cosine value is calculated using the cosine formula. The resulting graph will be a smooth curve that oscillates between -1 and 1, since those were the maximum values found on the unit circle for the cosine function.

We will plot points starting at 0 radians, where the cosine value is 1. This point is plotted at (0, 1) because the independent variable is 0 and the dependent variable is 1. Next, we noticed that our cosine value is 0 at 𝛑/2 radians, making the point (𝛑/2, 0) an x-intercept. We saw that our minimum cosine value occurred at 𝛑 radians, so we would plot the point (𝛑, -1) on the coordinate plane. We continue using the points on the unit circle to plot our graph of the cosine function, and connect our points using a smooth curve. We can now see that the cosine function looks like this:

Image courtesy of Nagwa.

As we go around the unit circle, the cosine value changes from 1 to -1, passing through 0 at 𝛑/2 radians. The cosine curve reaches its maximum value of 1 at 0 radians and its minimum value of -1 at 𝛑 radians. And like the sine curve, the cosine function is periodic, meaning it repeats every 2𝛑 radians.