Introduction

Howdy! Today we will go over AP Precalculus topic 3.3, which covers the values of the sine and cosine functions. This is a fundamental topic for trigonometry, which will frequently come up both in this course as well as college calculus courses.

AP Course and Exam Description Description

The AP Course and Exam Description expects you to be able to do the following for the AP exam in May:

Deriving Values from the Unit Circle

By now, you are probably familiar with the unit circle from previous algebra and geometry classes. However, if you are not, let’s do a quick review.

The unit circle is simply a circle with a radius of $1$, which we use to determine the values of sine and cosine. For any point plotted on the unit circle at an angle $\theta$ from the positive $x$-axis, the $x$-coordinate of the point gives us the cosine of the angle, and the $y$-coordinate gives us the sine of the angle. For reference, the unit circle is shown below.

Remember that all angles are now measured in radians, not degrees. Degrees are useful for cooking (and maybe physics), while radians are used in calculus and precalculus. In fact, the exam will remind you of this distinction.

Remember that all angles are now given in radians, NOT degrees. Degrees are for cooking (and maybe physics); radians are for calculus (and precalculus). In fact, the exam will remind you of this fact.

Now, we will plot all of these key values from the unit circle into a table for both cosine and sine from $x=0$ to $x=\pi$

$\cos (x)$

$\begin{array}{cccccccccccccccc}\mathbf{\theta }& 0& \frac{\pi }{6}& \frac{\pi }{4}& \frac{\pi }{3}& \frac{\pi }{2}& \frac{2\pi }{3}& \frac{3\pi }{4}& \frac{5\pi }{6}& \pi & \frac{7\pi }{6}& \frac{5\pi }{4}& \frac{5\pi }{3}& \frac{3\pi }{2}& \frac{7\pi }{4}& \frac{11\pi }{6}\\ \cos \mathbf{\theta }& 1& \frac{\sqrt{3}}{2}& \frac{\sqrt{2}}{2}& \frac{1}{2}& 0& -\frac{1}{2}& -\frac{\sqrt{2}}{2}& -\frac{\sqrt{3}}{2}& -1& -\frac{\sqrt{3}}{2}& -\frac{\sqrt{2}}{2}& \frac{1}{2}& 0& \frac{\sqrt{2}}{2}& \frac{\sqrt{3}}{2}\end{array}$

$\sin (x)$ 

$\begin{array}{cccccccccccccccc}\mathbf{\theta }& 0& \frac{\pi }{6}& \frac{\pi }{4}& \frac{\pi }{3}& \frac{\pi }{2}& \frac{2\pi }{3}& \frac{3\pi }{4}& \frac{5\pi }{6}& \pi & \frac{7\pi }{6}& \frac{5\pi }{4}& \frac{5\pi }{3}& \frac{3\pi }{2}& \frac{7\pi }{4}& \frac{11\pi }{6}\\ \cos \mathbf{\theta }& 1& \frac{\sqrt{3}}{2}& \frac{\sqrt{2}}{2}& \frac{1}{2}& 0& -\frac{1}{2}& -\frac{\sqrt{2}}{2}& -\frac{\sqrt{3}}{2}& -1& -\frac{\sqrt{3}}{2}& -\frac{\sqrt{2}}{2}& \frac{1}{2}& 0& \frac{\sqrt{2}}{2}& \frac{\sqrt{3}}{2}\end{array}$

We will see how the graph of both the cosine and the sine functions look like in the article for the next topic, which is topic 3.4. However, let’s just focus on the values of the function for now.

In the tables, I only gave the values of sine and cosine for the interval $0\le x\le 2\pi$. However, you may sometimes be asked about other intervals, such as $\pi \le x\le 2\pi$. Well, we can use the properties of symmetry of the unit circle to derive the values for those intervals as follows:

For angles between $\frac{\pi }{2}$ and $\frac{3\pi }{2}$, you may easily reflect the cosine values, or the right half of the unit circle.

For angles between $\pi$ and $2\pi$, you may reflect the sine values, or the top half of the unit circle

Practice Questions