<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# 4.3: What’s the Difference?

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Trigonometry, Chapter 3, Lesson 5.

## Problem 1 – Exploring the Angle Difference Formula for Cosine

Open the Cabri Jr. file called UNITCIRC. Observe a unit circle with points A\begin{align*}A\end{align*}, B\begin{align*}B\end{align*}, and C\begin{align*}C\end{align*} on the circle. Point O\begin{align*}O\end{align*} is the origin and center of the circle. The central angle AOB\begin{align*}\angle AOB\end{align*} represents the difference between AOC\begin{align*}\angle AOC\end{align*} and BOC\begin{align*}\angle BOC\end{align*}.

Move points A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} to see the changes to the on-screen measurements. To select an object, press the ALPHA key.

1. The x\begin{align*}x-\end{align*}coordinate of every ordered pair of a point on the unit circle represents the ________________________ of the corresponding angle.

2. The y\begin{align*}y-\end{align*}coordinate of every ordered pair of a point on the unit circle represents the ________________________ of the corresponding angle.

Use the file UNITCIRC to answer the following questions.

3. What is the sine of AOC\begin{align*}\angle AOC\end{align*} when its measure is about 100\begin{align*}100^\circ\end{align*}?

4. What is the cosine of AOC\begin{align*}\angle AOC\end{align*} when its measure is about 100\begin{align*}100^\circ\end{align*}?

5. What is the sine of BOC\begin{align*}\angle BOC\end{align*} when its measure is about 20\begin{align*}20^\circ\end{align*}?

6. What is the cosine of BOC\begin{align*}\angle BOC\end{align*} when its measure is about 20\begin{align*}20^\circ\end{align*}?

7. What is the sine of AOCBOC\begin{align*}\angle AOC - \angle BOC\end{align*} when mAOC=100\begin{align*}m \angle AOC = 100^\circ\end{align*} and mBOC=20\begin{align*}m \angle BOC = 20^\circ\end{align*}? Use the Cabri Jr. file to obtain your solution.

8. What is the cosine of AOCBOC\begin{align*}\angle AOC - \angle BOC\end{align*} when mAOC=100\begin{align*}m \angle AOC = 100^\circ\end{align*} and mBOC=20\begin{align*}m \angle BOC = 20^\circ\end{align*}? Use the Cabri Jr. file to obtain your solution.

9. Do you think the relationship between the values of sine and cosine for AOCBOC\begin{align*}\angle AOC -\angle BOC\end{align*} is quickly and easily obtained from the two individual angles as shown on the opening diagram for this activity? Explain you answer.

## Problem 2 – Applying the Angle Difference Formula

The opening diagram for this activity is commonly used in the derivation of the angle difference formula for cosine.

cos(AB)=cosAcosB+sinAsinB\begin{align*}\cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B\end{align*}

This formula is useful in finding exact values for the cosine of angles other than those you may already know from the unit circle.

For each exercise below, use your graphing calculator first with and without the formula. Then use the UNITCIRC circle graph.

10. Find the value of cos15\begin{align*}\cos 15^\circ\end{align*} by finding cos(6045)\begin{align*}\cos(60^\circ - 45^\circ)\end{align*}.

11. Find the value of cos75\begin{align*}\cos 75^\circ\end{align*} by finding cos(12045)\begin{align*}\cos(120^\circ - 45^\circ)\end{align*}.

12. Find the value of cos105\begin{align*}\cos 105^\circ\end{align*} by finding cos(??)\begin{align*}\cos(? - ?)\end{align*}. You choose the angles! Choose values that you recall from the unit circle.

## Extension 1 – Derivation of the Angle Difference Formula for Cosine

The Cabri Jr. file that has been used for this activity will be used in the derivation of the angle difference formula for cosine. As you look at the sketches below, find the angle represented by αβ\begin{align*}\alpha - \beta\end{align*}. Points A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} may be moved to change the measures of angles in the diagram.

The angle difference formula for cosine will be derived using the diagrams below.

13. Apply the Law of Cosines to the figure to the right to find an equation representing AB2\begin{align*}AB^2\end{align*}.

14. Apply the distance formula to the figure to the right to find an equation representing AB2\begin{align*}AB^2\end{align*}.

15. Combine the two equations obtained in Exercises 13 and 14 by setting them equal to each other. Solve for cos(αβ)\begin{align*}\cos(\alpha - \beta)\end{align*}. Test your resulting equation by entering values of your choice in UNITCIRC. Does your result agree with the directly calculated angle difference value? If not, check for algebraic and calculator entry errors.

## Extension 2 – Derivation of the Angle Sum Formula for Cosine

16. Now substitute β\begin{align*}-\beta\end{align*} in place of β\begin{align*}\beta\end{align*} into the angle difference formula for cosine and simplify the resulting equation. Test your resulting equation by entering values of your choice on page 3.3. Does your result agree with the provided angle sum value? If not, check for algebraic and calculator entry errors.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Show Hide Details
Description
Tags:
Subjects: