Lesson 4

The Unit Circle (Part 2)

  • Let’s look at angles and points on the unit circle.

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.

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.

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.

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:

\(2\pi\)

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:

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

(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.
(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 \(O\).
(From Unit 6, Lesson 3.)