Lesson 9 Water Wheels and the Unit Circle Practice Understanding

Ready

State a negative angle of rotation that is coterminal with the given angle of rotation. (Coterminal angles share the same terminal side of an angle of rotation.) Sketch and label both angles.

1.

Given $θ = \frac{π}{9}$

Coterminal angle:

2.

Given $θ = 1.66$

Coterminal angle:

3.

Given $θ = \frac{5 π}{4}$

Coterminal angle:

4.

Given $θ = \frac{3 π}{2}$

Coterminal angle:

5.

Given $θ = \frac{5 π}{3}$

Coterminal angle:

6.

What is the sum of a positive angle of rotation and the absolute value of its negative coterminal angle?

7.

Every angle has an infinite number of coterminal angles both positive and negative if the definition is extended to angles of rotation greater than $2 π$ . For example: an angle of $\frac{π}{4}$ is coterminal with angles of rotation measuring $\frac{9 π}{4}$ , $\frac{17 π}{4}$ , etc.

a.

Given $θ = \frac{2 π}{7}$ , name $2 positive coterminal angles$ .

b.

Given $θ = 5.13$ , name $2 positive coterminal angles$ .

Set

8.

Triangle $A B C$ is a right triangle. The length of one side is given. Fill in the values for the missing sides $m ∠ B = 30 °$ .

9.

Each point on the circle marks the end of a terminal ray in standard position. Label the measure of the angle of rotation at each position of the terminal ray. Angles will be in radians. Leave $π$ in your answer. (Each section is equal.)

10.

Use the values in #8 to write the exact coordinates of the points on the given circle. Be mindful of the numbers that are negative.

11.

Find the arc length, $s$ , from the point $(1, 0)$ to each point around the circle. Record your answers as decimal approximations to the nearest thousandth.

Use your calculator to find the following values.

12.

$\sin \frac{5 π}{6} =$

13.

$\sin \frac{π}{3} =$

14.

Why are both of your answers positive?

15.

$\cos \frac{2 π}{3} =$

16.

$\cos \frac{4 π}{3} =$

17.

Why are both of your answers negative?

18.

$\sin \frac{π}{2} =$

19.

$\cos \frac{π}{2} =$

20.

In which quadrant(s) are sine and cosine both negative?

21.

Name an angle of rotation in radians where sine is equal to $- 1$ .

22.

Name an angle of rotation where cosine is equal to $- 1$ .

Go

People in the 1950s and 1960s probably enjoyed music just as much as you do, but they didn’t have modern technology. They had vinyl records and phonographs. Vinyl records came in $3$ speeds. A record could be a $45$ , $33 \frac{1}{3}$ , or $78$ . These numbers referred to the rpms or revolutions per minute. $45 rpms$ means it would make $45 full circles$ in one minute.

23.

Calculate the angular speed of a $45 rpm$ , $33 \frac{1}{3} rpm$ , and $78 rpm$ record in degrees per minute.

a.

$45 rpm$

b.

$33 \frac{1}{3} rpm$

c.

$78 rpm$

Angular speed describes how fast something is turning. Linear speed describes how far it travels while it is turning. Linear speed depends on the circumference of a circle $(C = 2 π r)$ and the number of revolutions per minute.

Vinyl records were not the same size. A $45$ rpm record had a diameter of $7 inches$ , a $33 \frac{1}{3}$ a diameter of $12 inches$ , and a $78$ had a diameter of $10 inches$ .

24.

a.

If a fly landed on the outer edge of a $45 rpm$ record, how far would it travel in $1 minute$ ?

b.

How far if it was perched on the outside edge of a $33 \frac{1}{3} rpm$ record?

c.

How far if it was perched on the outside edge of a $78 rpm$ record?