Trig Functions and the Unit Circle

Task

The points on the graphs and the unit circle below were chosen so that there is a relationship between them.

Explain the relationship between the coordinates $a,\ b,\ c,\ $ and $d$ marked on the graph of $y=\sin t$ and $y=\cos t$ and the quantities $A,\ B,$ and $C$ marked in the diagram of the unit circle below.

IM Commentary

The purpose of this task is to help students make the connection between the graphs of $\sin t$ and $\cos t$ and the $x$ and $y$ coordinates of points moving around the unit circle. Students have to match coordinates of points on the graph with coordinates and angles in the diagram of the unit circle.

A slight variation of this task would be to ask the students to draw in line segments in both diagrams that correspond to the indicated quantities. So for example, $C$ would be the line segment along the $t$ axis of the graph from $(0,0)$ to $(0,a)$.

In this task students practice SMP 7 - Look for and Make Use of Structure. A firm understanding of the connections between the unit circle and the graphs of sine and cosine build a solid foundation for future work in trigonometry.

Solution

On the graph, $a$ and $c$ are the input value of the sine and cosine function, respectively, that give the output values $b=\sin(a)$ and $d=\cos(c)$.

On the unit circle, we know that $A=\cos(C)$ and $B=\sin(C).$

We can estimate that $a=c\approx 2.1.$ We can also estimate that $C$ is an angle close to $\pi/3\approx 2.1.$ We are given that the points on the graph and the unit circle correspond to each other, and therefore we can conclude that $a=c=C.$

Putting everything together we get $b=\sin(a)=\sin(C)=B$ and $d=\cos(c)=\cos(C)=A.$