1.2 - Introduction to the Unit Circle

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Objectives:

• Convert between radians and degrees.
• Fill in the unit circle.

Definitions

the unit circle, radians

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Lesson Guide

Introduction to Radians

Start by introducing students to the idea of the unit circle. They have a blank copy of the unit circle in their worksheet (Problem 6). Let them know that the goal for the class will be to fill in the details of the unit circle: the angles as well as the coordinates of the points.

 The $\underline{unit circle}$ is a circle of radius 1, centered at the origin. When measuring an angle around the unit circle, 
 we travel in the counter-clockwise direction, starting from the positive $x$-axis. A negative angle is measured in the opposite, 
 or clockwise, direction. A complete trip around the unit circle amounts to a total of 360 degrees. 

Have students fill in the degree measures of the angles marked on the unit circle, given the following information: 12 of the angles are obtained by moving $360/12=30^{\circ}$ at a time around the unit circle, and the other 4 are obtained by bisecting each of the quadrants (you may have to remind students that each quadrant constitutes $90^{\circ}$.

Introduce students to the idea of using radians as an alternative angle measure. One option is to tell students that radians are related to the circumference of the unit circle. Ask them what the circumference of the unit circle is, and then tell them that a complete trip around the unit circle is $2\pi$ radians:

 Radians arise from looking at angles as a fraction of the circumference of the unit circle; a complete trip around the unit circle 
 amounts to a total of $2\pi$ radians.
 *Even though radians are related to the circumference of the {\bf unit} circle, radians are a unit for measuring angles. 
 Make sure students are aware of this distinction.

Have students fill in the radian measures of each of the angles on their unit circles. You may want to give a couple of examples converting degrees to radians before having them do this, or you may let them explore this relationship on their own.

Solidify the relationship between degrees and radians. You can use the unit circle as a jumping-off point, and then do an example or two converting back and forth with angles that are not on their unit circles. Tell students that when working with physical quantities (angle above a horizon, interior angle of a triangle, etc.), we tend to work with degrees. On the other hand, when working in abstract settings (graphing functions, solving equations, etc.), we tend to work with radians. Our default angle measurements will be made in radians. Unless degrees are specified for a particular problem, students should assume that all angle measures are in radians.

Have students complete Problems 1-5 on the worksheet. Regroup to make sure that everyone has arrived at the correct conclusions in Problem 3.

Filling in the Unit Circle

Work on filling in the coordinates of the points in the the first quadrant of the unit circle. How much detail you go into with the derivations is up to you, but students will need to have at least the first quadrant of their unit circles completed by the end of class. Below are some examples of how you might present the derivations:

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Let's identify the $(x,y)$-coordinates of the point on the unit circle corresponding to an angle of $45^\circ$ or $\pi/4$ radians. Consider the following illustration:

• Since we are working on the unit circle, our radius is 1 unit, so $r = 1$.
• Recall that the Pythagorean theorem states that, for right triangles \textit{only}, $a^2+b^2=c^2,$ where $a$ and $b$ represent the lengths of the triangle's legs, and $c$ represents the length of the hypotenuse. How can we rewrite this using the variables from our illustration?
• Since $\theta = \pi/4$, $y = x$. We can use this, along with $r=1$, to solve for $x$: $x=\frac{\sqrt{2}}{2}$.
• We conclude that $y = \frac{\sqrt{2}}{2}$ as well.
• Have students fill in these values on their unit circles.

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Now let's identify the $(x,y)$ point which corresponds to an angle of $\pi/6$ radians or 30 degrees on the unit circle. Consider the following illustration:

• The \emph{large} triangle shown is an equilateral triangle since $r$ and $s$ are equal (why?) and one of the angles is $60^\circ$ (which angle?). So we know that the $y$-coordinate of the point $P$ is $y=1/2$.
• Now we can use the Pythagorean theorem again to find $x$: $r^2 = x^2 + y^2$, so $x = \frac{\sqrt{3}}{2}$.
• Have students fill in these values on their unit circles.
• From here, you could have students follow a similar process to derive the coordinates for the angle $\pi/3$.

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Have students use what they know about the coordinates in the first quadrant to fill out the remainder of the unit circle. Walk among groups to ensure that everyone arrives at a consensus. Encourage students to commit the unit circle to memory.

Comments

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In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

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One way to explain the relationship between degrees and radians is by comparing them to using feet vs. meters when measuring length or gallons vs. liters when measuring the volume of a liquid. Both degrees and radians measure the same thing - angles, but are different units of measurement.

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There are two possible answers to Problem 5 in the workbook. You might want to observe if any of your students notice this while they are working on this problem.

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