Lesson 3

The Unit Circle (Part 1)

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Problem 1

\(C\) is a circle with radius \(r\). Which of the following is true? Select all that apply.

A:

The diameter of \(C\) is \(2r\).

B:

The circumference of \(C\) is \(\pi r\).

C:

The circumference of \(C\) is \(2\pi r\).

D:

One quarter of the circle has length \(\frac{\pi r}{4}\).

E:

One quarter of the circle has length \(\frac{\pi r}{2}\).

Solution

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Problem 2

angle measure rotation
0 0
\(\frac{\pi}{6}\)  
  \(\frac{1}{8}\)
  \(\frac{1}{6}\)
\(\frac{\pi}{2}\)  
\(\frac{2\pi}{3}\)  
  \(\frac{1}{2}\)
\(\frac{3\pi}{2}\)  
  \(\frac{7}{8}\)
  1

The table shows an angle measure in radians and the amount of rotation about a circle corresponding to the angle. For example, \(2\pi\) radians corresponds to 1 full rotation. Complete the table.

Solution

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Problem 3

A wheel has a radius of 1 foot. After the wheel has traveled a certain distance in the counterclockwise direction, the point \(P\) has returned to its original position. How many feet could the wheel have traveled? Select all that apply.

Circle, partitioned by 12 congruent central angles. Radius 1 foot. Point P lies on the circle at the right end of the horizontal diameter.
A:

\(\frac{\pi}{2}\)

B:

\(\pi\)

C:

\(2\pi\)

D:

\(5\pi\)

E:

\(10\pi\)

Solution

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Problem 4

Here are some points labeled on the unit circle:

Unit circle inscribed in a coordinate plane. Point 1 comma 0 is labeled P, point 0 comma 1 is labeled R. Q is a point on the circle between P and R.
  1. What is the measure in radians of angle \(POR\)?
  2. Angle \(POQ\) is halfway between 0 radians and angle \(POR\). What is the measure in radians of angle \(POQ\)?
  3. Label point \(U\) on the circle so that the measure of angle \(POU\) is \(\frac{3\pi}{4}\).
  4. Label point \(V\) on the circle so that the measure of angle \(POV\) is \(\frac{3\pi}{2}\).

Solution

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Problem 5

  1. Mark the points on the unit circle with \(x\)-coordinate \(\frac{4}{5}\).

    A circle with radius 1 inscribed on a coordinate plane, centered at the origin.
  2. What are the \(y\)-coordinates of those points? Explain how you know.

Solution

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Problem 6

The point \((8, 15)\) lies on a circle centered at \((0,0)\). Where does the circle intersect the \(x\)-axis? Where does the circle intersect the \(y\)-axis? Explain how you know.

Solution

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(From Unit 6, Lesson 1.)

Problem 7

Triangles \(ABC\) and \(DEF\) are similar. Explain why \(\tan(A) = \tan(D)\).

Two right triangles. First, A, B C with opposite sides labeled lower case a, b and c. Second, smaller triangle E D F with opposite sides labeled lower case e d f. Angles C and F are right.

Solution

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(From Unit 6, Lesson 2.)