2.2: Round and Round She Goes
This activity is intended to supplement Trigonometry, Chapter 1, Lesson 7.
ID: 12385
Time Required: 20 minutes
Activity Overview
In this activity, students will explore relationships on the unit circle. Students will identify coordinates of points given an angle measure in degrees.
Topic: Unit Circle
- Right triangle trigonometry and the unit circle
- Special right triangles
- Cosine and sine on the unit circle
Teacher Preparation and Notes
- The first problem and second problem engage students in the exploration of the connection between angle measure and the coordinates of points in the first quadrant.
Associated Materials
- Student Worksheet: Round and Round She Goes http://www.ck12.org/flexr/chapter/9699, scroll down to the second activity.
Problem 1 – Introduction to the Unit Circle
Students are introduced to the concept of the unit circle. Right triangle relationships are explored to develop an understanding of the patterns involved.
Special right triangles are addressed to help students understand the exact values they will likely be expected to know.
Students often have difficulty with remembering some of these special values. Ask students if they can see a pattern that might help them remember that \begin{align*}\frac{\sqrt{2}}{2}\end{align*} goes with the \begin{align*}45-45-90\end{align*} triangle. Students may point out that there is a pattern of twos—two angles and two sides are the same.
Similarly, ask them how they might remember that \begin{align*}\frac{\sqrt{3}}{2}\end{align*} goes with the \begin{align*}30-60-90\end{align*} triangle. Students may state that there is a pattern of threes—all angles involved are multiples of 3. Next, once students realize that \begin{align*}\frac{\sqrt{3}}{2}\end{align*} and \begin{align*}\frac{1}{2}\end{align*} go together in an ordered pair, it is easy to point out using a unit circle sketch that a \begin{align*}60^\circ \end{align*} angle involves an \begin{align*}x-\end{align*}value that is \begin{align*}\frac{1}{2}\end{align*} and the \begin{align*}30^\circ \end{align*} angle involves a \begin{align*}y-\end{align*}value that is \begin{align*}\frac{1}{2}\end{align*}.
Problem 2 – Extending the Pattern
Students use a visual model to extend what they established in Quadrant I to Quadrants II, III, and IV.
It is very helpful for students to think about symmetry as they move on to these other quadrants. Construction of rectangles in the unit circle is helpful for many students to make this extension.
Solutions
- \begin{align*}x = \cos \theta\end{align*}
- \begin{align*}x = \sin \theta \end{align*}
- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{\sqrt{2}}{2}\end{align*}
- \begin{align*}\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)\end{align*}
- \begin{align*}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\end{align*}
- \begin{align*}\frac{\sqrt{3}}{2}\end{align*}
- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{1}{2}\end{align*}
- \begin{align*}\frac{\sqrt{3}}{2}\end{align*}
- \begin{align*}\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\end{align*}
- \begin{align*}\frac{\sqrt{2}}{2}\end{align*}
- \begin{align*}\frac{\sqrt{2}}{2}\end{align*}
- \begin{align*}(-a, b)\end{align*}
- \begin{align*}(-a, -b)\end{align*}
- \begin{align*}(a, -b)\end{align*}
- \begin{align*}120^\circ\end{align*}
- \begin{align*}240^\circ\end{align*}
- \begin{align*}300^\circ \end{align*}
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