Lesson 7 Staking It Solidify Understanding

Learning Focus

Calculate arc length for angles of rotation measured in radians.

Visualize the size of angles measured in radians, including radians given in decimal form.

How large is a radian? How can I estimate the size of angles measured in radians relative to degrees?

What good are radians? Are any calculations made easier when the angle is measured in radians? Are any contexts easier to describe using radians?

Open Up the Math: Launch, Explore, Discuss

After considering different plans for laying out the archeological site described in Diggin’ It, Alyce, Javier, and Veronica have decided to make concentric circles at $10 meter$ intervals from the central tower. They have also decided to use $16 stakes$ per circle, in order to have a few more points of reference. Using ropes of different lengths to keep the radius constant, they have traced out these circles in the sand. Because they know the circles will soon be worn away by the wind and by people’s footprints, they feel a sense of urgency to locate the positions of the $16 stakes$ that will mark each circle. The team wants to be efficient and make as few measurements as possible.

1.

Veronica suggests they should locate the stakes around one circle and use those positions to mark where the stakes will go on all of the other circles. What do you think about Veronica’s idea?

a.

How will marking stake positions on one circle help them locate the positions of the stakes on all of the other circles?

b.

If there are $16$ stakes per circle, how far apart are the stakes in degrees?

c.

How far apart are the stakes in radians?

Veronica has decided they should stake out the circle with a radius of $50 meters$ first. She is standing at the point $(50, 0)$ and knows she needs to move $22 \frac{1}{2} °$ around the circle to place her next stake. But, she wonders, “How far is that?”

Veronica decides she will find the distance by setting up a proportion using degree measurements.
Alyce thinks they should find the distance by taking $\frac{1}{16}$ of the circumference.
Javier thinks they should use radian measurement in their calculation.

2.

Show how each team member will calculate this distance.

Veronica’s strategy:

Alyce’s strategy:

Javier’s strategy:

Javier has a different idea. He suggests they should figure out the locations of all of the stakes in Quadrant I first, and then it would be easy to find the locations of the stakes in all the other quadrants by using the Quadrant I locations.

3.

What do you think about Javier’s suggestion? How will marking the location of stakes in Quadrant I help them figure out the location of the stakes in other quadrants?

Javier has already started working on his strategy and has completed the calculations for several random points in Quadrant I (see Javier’s diagram).

Alyce has a concern: “Using the $(x, y)$ coordinates means we have to start at the central tower and move a horizontal distance, then a vertical distance, and then drive in the stake. Then we have to go back to the central tower and start over again to locate another stake. I just want to walk around the circle and drive a stake at equal intervals. I think we need the arc length, not the rectangular coordinates of the points.”

Veronica has another idea: “I want to use a compass to orient the direction I should walk from the central tower, and then drive a stake every $10 meters$ along that line. Then I can return to the central tower and head off in a different direction. Unfortunately, my compass gives me angles in degrees, and Javier labeled his diagram in radians. I need to label each point using $(r, θ)$ coordinates.”

4.

Develop a strategy to locate all of the other stakes in the first quadrant for these additional circles. Find the $(x, y)$ coordinates to complete Javier’s strategy, the $(r, θ)$ coordinates for Veronica’s strategy, and the arc lengths from the positive $x$ -axis to each point for Alyce’s strategy.

5.

Javier has noticed that the farther away from the central tower, the farther apart the stakes appear to be. He suggests they should add radial lines to the diagram at angles of $\frac{π}{16}, \frac{3 π}{16}, \frac{5 π}{16}, and \frac{7 π}{16}$ , but only locate stakes at these angles on the circle of radius $40 meters$ . What do these angles measure in degrees?

6.

Alyce and Veronica love this idea and assign Javier the job of posting all of the stakes on the $40 meter$ circle, since there will be twice as many of them. Javier realizes he doesn’t need to calculate the rectangular coordinates for each point to locate their positions. He plans to just walk around the circle, posting stakes at equal intervals. How far apart are the stakes on the $40 meter$ circle? Calculate the arc length using Javier’s method from problem 2.

Pause and Reflect

7.

Being able to convert quickly between degree measurement and radian measurement will help Javier a lot.

a.

Develop a strategy Javier can use consistently to convert angle measurements from degrees to radians.

b.

Develop a strategy Javier can use consistently to convert angle measurements from radians to degrees.

8.

Veronica has a different proposal for the soon-to-be-added $50 meter$ circle. She wants to place a stake every $15$ °.

a.

Show how to calculate the arc length between stakes using $15 °$ in your computation.

b.

Show how to calculate the arc length between stakes using radian measure.

Ready for More?

Explain how the radian measure shows up in Alyce’s, Javier’s, and Veronica’s strategies for problem 2.

Veronica’s strategy:

Alyce’s strategy:

Javier’s strategy:

Takeaways

If a circle has been divided into $n$ equal arcs, then the angle measure of one arc is or radians.

I can find arc length for an angle of rotation measured in , by using the formula .

I can convert degree measurement to radians by .

I can convert radian measurement to degrees by .

Vocabulary

arc length
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we continued to work with degree and radian measurement for angles of rotation. We found strategies for converting from one angle measurement to the other, and we saw that the formula for finding arc length for angles measured in radians was simpler than the formula for finding arc length for angles measured in degrees. This occurred because radian measure is defined as a ratio of arc length to radius.

Retrieval

1.

Find one point in each quadrant that lies on the circle $x^{2} + y^{2} = 16$ .

a.

Quadrant I:

b.

Quadrant II:

c.

Quadrant III:

d.

Quadrant IV:

2.

Label each point on the circle provided with the measure of the angle of rotation in standard position beginning with the rotation from $H$ to $A$ . Angle measures should be in radians. (Recall a full rotation around the circle would be $2 π radians$ .)

Point $A$ :

Point $B$ :

Point $C$ :

Point $D$ :

Point $E$ :

Point $F$ :

Point $G$ :

Point $H$ :