1.3 - The Sine & Cosine Functions

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Objectives:

• Understand the purpose of the sine and cosine functions.
• Understand the relationship between sine and cosine and right triangles.
• Use the unit circle to find the sine and cosine of an angle.
• Use the Pythagorean identity.

Definitions

sine, cosine

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Lesson Guide

The Unit Circle

 Given an angle $\theta$ (in either degrees or radians), and the $(x,y)$-coordinates of the corresponding point 
 on the unit circle, we define $\cos(\theta)=x$, and $\sin(\theta)=y$.
 *Note that sine and cosine are functions which take angles as inputs.

Give several examples from the unit circle, which students should have completely filled out, to emphasize the relationship between sine and cosine and coordinates on the unit circle.

SOH-CAH-TOA

Students may have also seen sine and cosine defined in the context of right triangles, which you should also introduce:

Where $\sin(\theta)=\frac{opposite}{hypotenuse}=o/h$, and $\cos(\theta)=\frac{adjacent}{hypotenuse}=a/h$.

This gives rise to the popular mnemonic SOH-CAH-TOA, where the first two abbreviations stand for "sine is opposite over hypotenuse," and "cosine is adjacent over hypotenuse." (TOA relates to another function which we'll look at later.)

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Have students find the sine and cosine of an angle in an example right triangle, such as the one below:

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Emphasize to students that the two definitions presented are the same! Point out that on the unit circle, the length of the hypotenuse is equal to 1, so the definitions match.

Have students work through Problems 1 and 2 on the worksheet. Students should first try to find the sine and cosine of an angle by using their unit circles. Only if the angles aren't on the unit circle should they use their calculators. Make sure students know how to switch back and forth between degree and radian mode on their calculators.

Graphs of Sine and Cosine

Introduce the graphs of the sine and cosine functions (make sure to emphasize that inputs are in radians). You may choose to plot a few values from the unit circle first, and then connect them. This is a good chance to emphasize the relationship between sine and cosine and coordinates on the unit circle.

Have students work through Problem 5 on their worksheet, then discuss their observations as a whole class.

Have students complete the worksheet. Note that Problem 6 deals with the Pythagorean Identity so you should be sure to go over this concept with your class. Part 6a may take students quite some time. Best to have particular tables make certain calculation and have them report back to the class.

Problems 7 and 8 deal with finding the coordinates of a point on a circle of radius larger than 1 and circles not centered at the origin.

 At some point during lecture, you should tell students that to represent a trigonometric function raised to a power, for example 
 $(\sin(x))^2$, we write $\sin^2(x)$, rather than $\sin(x)^2$. A good place for this is before Problem 6.

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.

- Some students seemed to get tripped up with relating the (x,y) = (r cos(theta), r sin(theta)) to right triangles when the triangles were not in the first quadrant. They thought "Shouldn't I be calculating sin and cos using the angle be *inside* the triangle (instead of the angle from the x-axis)?" So maybe doing an example like this would be helpful.