Lesson 4

The Unit Circle (Part 2)

View Student Lesson

Problem 1

Angle \(ABC\) measures \(\frac{\pi}{3}\) radians, and the coordinates of \(C\) are about \((0.5,0.87)\).

A circle with center B at the origin of an x y plane.
  1. The measure of angle \(ABD\) is \(\frac{2\pi}{3}\) radians. What are the approximate coordinates of \(D\)? Explain how you know.
  2. The measure of angle \(ABE\) is \(\frac{5\pi}{3}\) radians. What are the approximate coordinates of \(E\)? Explain how you know.

Solution

For access, consult one of our IM Certified Partners.

Problem 2

Give an angle of rotation centered at the origin that sends point \(P\) to a location whose \((x,y)\) coordinates satisfy the given conditions.

  1. \(x > 0\) and \(y < 0\)
  2. \(x < 0\) and \(y > 0\)
  3. \(y < 0\) and \(x < 0\)
Circle on a coordinate plane, center at the origin, radius 1. Point P at 1 comma 0.

Solution

For access, consult one of our IM Certified Partners.

Problem 3

Lin calculates \(0.97^2 + 0.26^2\) and finds that it is 1.0085. 

  1. Explain why \((0.97,0.26)\) is not on the unit circle. 
  2. Is \((0.97,0.26)\) a good estimate for the coordinates of a point on the unit circle? Explain how you know.

Solution

For access, consult one of our IM Certified Partners.

Problem 4

The \(x\)-coordinate of a point \(P\) on the unit circle is 0. If point \(P\) is the result of rotating the point \((1,0)\) by \(\theta\) radians counterclockwise about the origin, what angle could \(\theta\) represent? Select all that apply.

A:

0

B:

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

C:

\(\pi\)

D:

\(\frac{3\pi}{2}\)

E:

\(2\pi\)

Solution

For access, consult one of our IM Certified Partners.

Problem 5

Here is triangle \(ABC\). \(BC\) is shorter than \(AC\). Which statements are true? Select all that apply.

Triangle A, B C. Angle C is a right angle.
A:

\(\sin(A) > 1\)

B:

\(\tan(A) < 1\)

C:

\(\cos(A) < 1\)

D:

\(\sin(A) < \sin(B)\)

E:

\(\cos(A) < \cos(B)\)

F:

\(\tan(A) < \tan(B)\)

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 2.)

Problem 6

Angle \(POQ\) measures one radian. The radius of the circle is 1 unit.

  1. What is the length of arc \(PQ\)?
  2. Explain why the length of arc \(PQ\) is less than \(\frac{1}{6}\) of the full circle.
Unit circle inscribed in a coordinate plane. Point 1 comma 0 is labeled P, Q is a point on the circle in the first quadrant.

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 3.)

Problem 7

Label these points on the unit circle:

Unit circle inscribed in a coordinate plane. Point 1 comma 0 is labeled P.
  1. \(Q\) is the image of \(P\) after a \(\frac{11\pi}{6}\) rotation with center \(O\).
  2. \(R\) is the image of \(P\) after a \(\frac{3\pi}{2}\) rotation with center \(O\).
  3. \(U\) is the image of \(P\) after a \(\frac{2\pi}{3}\) rotation with center \(O\).
  4. \(V\) is the image of \(P\) after a \(\frac{\pi}{3}\) rotation with center \(O\).

Solution

For access, consult one of our IM Certified Partners.

(From Unit 6, Lesson 3.)