# 2.3: Trigonometric Patterns

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 1, Lesson 8.

ID: 12434

Time required: 15 minutes

## Activity Overview

Students will use the unit circle to examine patterns in the six trigonometric functions.

Topic: Trigonometry

• Unit Circle
• Right Triangle Trigonometry

Teacher Preparation and Notes

Associated Materials

## Trigonometric Patterns

Students will move the triangle on the unit circle to find the angle measures listed in the table on the student worksheet. Students will record the values and then answer questions about patterns in the results. Because the Cabri Jr. file only measures the angle less than \begin{align*}90^\circ\end{align*}, there is opportunity for some further student learning. This means that when the angle being displayed is \begin{align*}30^\circ\end{align*} and the point is in the second quadrant, the angle being observed is really \begin{align*}150^\circ\end{align*} \begin{align*}(180^\circ - 30^\circ)\end{align*}.

Students can write the ratios on their worksheet and then use the Home screen to do their calculations.

Note that due to rounding in Cabri Jr., the answers are approximate, not exact.

Discussion Questions

• What happens at \begin{align*}90^\circ\end{align*}, \begin{align*}180^\circ\end{align*}, \begin{align*}270^\circ\end{align*}, and \begin{align*}360^\circ\end{align*}?
• Why is the tangent function undefined for some angle measures?

## Extension – Patterns in reciprocal functions

In this problem, students will repeat the activity for the co-trigonometric functions. Students should notice that these functions are the reciprocals of the functions from the first part of the activity by looking at the given formulas. This will help them in calculating these functions because they can use simply find the reciprocals on the Home screen instead of finding new ratios.

Discussion Questions

• Are any of the functions undefined? For what values?

Do any of the patterns match the patterns from the functions in the activity?

## Solutions

\begin{align*}\theta\end{align*} \begin{align*}\sin\theta \end{align*} \begin{align*}\cos\theta \end{align*} \begin{align*}\tan\theta \end{align*}
\begin{align*}30^\circ\end{align*} 0.5 0.866 0.577
\begin{align*}45^\circ\end{align*} 0.707 0.707 1
\begin{align*}60^\circ\end{align*} 0.866 0.5 1.732
\begin{align*}90^\circ\end{align*} 1 0 Undefined
\begin{align*}120^\circ\end{align*} 0.866 -0.5 -1.732
\begin{align*}135^\circ\end{align*} 0.707 -0.707 -1
\begin{align*}150^\circ\end{align*} 0.5 -0.866 -0.577
\begin{align*}180^\circ\end{align*} 0 -1 0
\begin{align*}210^\circ\end{align*} -0.5 -0.866 0.577
\begin{align*}225^\circ\end{align*} -0.707 -0.707 1
\begin{align*}240^\circ\end{align*} -0.866 -0.5 1.732
\begin{align*}270^\circ\end{align*} -1 0 Undefined
\begin{align*}300^\circ\end{align*} -0.866 0.5 -1.732
\begin{align*}315^\circ\end{align*} -0.707 0.707 -1
\begin{align*}330^\circ\end{align*} -0.5 0.866 -0.577
\begin{align*}360^\circ\end{align*} 0 1 0

1. \begin{align*}0^\circ < \theta < 180^\circ\end{align*}

2. \begin{align*}180^\circ < \theta < 270^\circ\end{align*}

3. Positive \begin{align*}0^\circ < \theta < 90^\circ, 180^\circ < \theta < 270^\circ\end{align*} because sine and cosine have same sign Negative \begin{align*}90° < \theta < 180^\circ, 270^\circ < \theta < 360^\circ\end{align*} because sine and cosine have different signs

4. \begin{align*}\cos(330^\circ)\end{align*}

5. Possible response: \begin{align*}\cos(45^\circ) = \cos(315^\circ); \cos(60^\circ) = \cos(300^\circ)\end{align*}

6. \begin{align*}\tan(225^\circ)\end{align*}

7. Possible response: \begin{align*}\tan(60^\circ) = \tan(240^\circ), \tan(30^\circ) = \tan(210^\circ)\end{align*}

8. Answers will vary. Sample response: The values in the first quadrant are repeated in the other quadrants, but have different signs.

9. Answers will vary. Sample response: The values of sine and cosine switch within a quadrant, such as \begin{align*}\sin(30^\circ) = \cos(60^\circ)\end{align*}, but the signs may be opposite in some quadrants.

Extension

\begin{align*}\theta\end{align*} \begin{align*}\sin \theta\end{align*} \begin{align*}\cos \theta\end{align*} \begin{align*}\tan \theta\end{align*}
\begin{align*}30^\circ\end{align*} 1.155 2 1.732
\begin{align*}45^\circ\end{align*} 1.414 1.414 1
\begin{align*}60^\circ\end{align*} 2 1.155 0.577
\begin{align*}90^\circ\end{align*} Undefined 1 0
\begin{align*}120^\circ\end{align*} -2 1.155 -1.732
\begin{align*}135^\circ\end{align*} -1.414 1.414 -1
\begin{align*}150^\circ\end{align*} -1.155 2 -1.732
\begin{align*}180^\circ\end{align*} -1 Undefined Undefined
\begin{align*}210^\circ\end{align*} -1.155 -2 1.732
\begin{align*}225^\circ\end{align*} -1.414 -1.414 1
\begin{align*}240^\circ\end{align*} -2 -1.155 0.577
\begin{align*}270^\circ\end{align*} Undefined -1 0
\begin{align*}300^\circ\end{align*} 2 -1.155 -0.577
\begin{align*}315^\circ\end{align*} 1.414 -1.414 -1
\begin{align*}330^\circ\end{align*} 1.144 -2 -1.732
\begin{align*}360^\circ\end{align*} 1 Undefined Undefined

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