Lesson 8 “Sine”ing and “Cosine”ing It Solidify Understanding

Jump Start

Notice and Wonder

Describe at least two things you notice and one thing you are wondering about as you examine the following four trigonometric functions.

1.

$f (x) = 2 \sin (3 x)$

2.

$g (θ) = 4 \cos (θ)$

3.

$h (t) = 3 \sin (\frac{π}{5} t)$

4.

$k (t) = 5 \cos (\frac{12 °}{second} \cdot t)$

Learning Focus

Find a relationship between the arc length and $(x, y)$ coordinates of a point on the circle of radius $1$ .

How does the unit circle make our work with trigonometric functions easier?

Open Up the Math: Launch, Explore, Discuss

In the previous lessons of this unit, you have used the similarity of circles, the symmetry of circles, right triangle trigonometry, and proportional reasoning to locate stakes on concentric circles. In this lesson, we consider points on the simplest circle of all, the circle with a radius of $1$ , which is often referred to as the unit circle.

1.

Here is a portion of a unit circle—the portion lying in the first quadrant of a coordinate grid. As in the previous lesson, Staking It, this portion of the unit circle has been divided into intervals measuring $\frac{π}{8}$ radians. As done previously, find the coordinates of each of the indicated points in the diagram. Also find the arc length, $s$ , from the point $(1, 0)$ to each of the indicated points.

DO NOT USE A CALCULATOR AS YOU LABEL THE COORDINATES OR ARC LENGTH. Instead, leave your results as fractions or as trigonometric expressions—exact values, rather than decimal approximations.

2.

Repeat the labeling of the points using a calculator and the work you did with this diagram previously. That is, the $x$ - and $y$ -coordinates and the arc length should be given as decimal approximations.

Pause and Reflect

Javier has been wondering if his calculator will allow him to calculate trigonometric values for angles measured in radians, rather than degrees. He feels like this will simplify much of his computational work when trying to locate the coordinates of stakes on the circles surrounding the central tower of the archeological site.

After consulting his calculator’s manual, Javier has learned that he can set his calculator in radian mode. Now he is ready to examine the following calculations.

3.

With your calculator set in radian mode, find each of the following values. Record your answers as decimal approximations to the nearest thousandth.

$\sin (\frac{π}{8}) =$	$\cos (\frac{π}{8}) =$	$\frac{π}{8} =$
$\sin (\frac{π}{4}) =$	$\cos (\frac{π}{4}) =$	$\frac{π}{4} =$
$\sin (\frac{3 π}{8}) =$	$\cos (\frac{3 π}{8}) =$	$\frac{3 π}{8} =$
$\sin (\frac{π}{2}) =$	$\cos (\frac{π}{2}) =$	$\frac{π}{2} =$

4.

The coordinates and arc lengths you found for points on the unit circle seem to be showing up in Javier’s computations. Why is that so? That is…

Explain why the radian measure of an angle on the unit circle is the same as the arc length.

Explain why the sine of an angle measured in radians is the same as the $y$ -coordinate of a point on the unit circle.

Explain why the cosine of an angle measured in radians is the same as the $x$ -coordinate of a point on the unit circle.

5.

Based on this work, find the following without using a calculator:

$\sin (\frac{5 π}{8}) =$

$\cos (\frac{7 π}{8}) =$

$\cos (π) =$

So, why are radians important enough to have their own mode on a calculator? Here are some things to think about.

6.

We have defined radians as the ratio of arc length to radius. If the arc length and radius are both measured in meters (or centimeters, or inches or feet), what are the units on the ratio?

7.

If we define radians as the arc length of an intercepted arc on the unit circle $x^{2} + y^{2} = 1$ , how does the length of an arc that measures $1 radian$ on the unit circle compare to the length of $1 unit$ on the $x$ -axis?

8.

Explain what this statement means: Radians are important because they measure angles using real numbers instead of degrees.

Since we can think of the real numbers along the $x$ -axis as the radian measure of an angle of rotation, we can graph sine and cosine functions as functions of $x$ . It helps to label the $x$ -axis in a way that highlights the real numbers that represent a full rotation, a half rotation, or some other useful fraction of a rotation. The following are two possible examples.

9.

Graph each of the following functions on one of the grids provided, one function per grid. Be prepared to explain your graphing strategy, and why you chose to use the grid you used for each graph.

$f (x) = 2 \sin (x)$
$g (x) = \sin (2 x)$

Ready for More?

Use the graph of the unit circle to reason about the following problems:

Where would the terminal ray of an angle that measures $2 radians$ be located?

Is the sine of an angle that measures $2 radians$ larger or smaller than the sine of an angle that measures $3 radians$ ?

A point whose coordinates can be approximated by $(0.5, 0.866)$ is on the unit circle. What is the sine of the angle whose terminal ray passes through this point?

About how many degrees do you think this angle of rotation would measure?

Takeaways

Every point on the unit circle gives the sine and cosine values of an angle of rotation because:

The arc length of an angle of rotation measures
The $x$ -coordinate of the point gives
The $y$ -coordinate of the point gives

The unit circle defines trigonometric functions over the domain of real numbers because:

Vocabulary

unit circle
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about the unit circle and how it models the sine and cosine values for every angle of rotation. We also found that the radian measure of an angle of rotation is represented by the arc length of the intercepted arc.

Retrieval

1.

Write an equivalent expression. Rationalize the denominators when appropriate.

a.

$\frac{2}{\sqrt{7}}$

b.

$\frac{\frac{\sqrt{3}}{2}}{\frac{5}{2}}$

2.

Change the given angles to radians.

a.

$80 °$

b.

$135 °$

3.

Change the given angles to degrees.

a.

$\frac{4 π}{3}$

b.

$\frac{17 π}{9}$