# 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 \((0.5,0.87)\).

- The measure of angle \(ABD\) is \(\frac{2\pi}{3}\) radians. What are the approximate coordinates of \(D\)? Explain how you know.
- 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.

- \(x > 0\) and \(y < 0\)
- \(x < 0\) and \(y > 0\)
- \(y < 0\) and \(x < 0\)

### Problem 3

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

- Explain why \((0.97,0.26)\) is not on the unit circle.
- 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.

0

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

\(\pi\)

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

\(2\pi\)

### Problem 5

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

\(\sin(A) > 1\)

\(\tan(A) < 1\)

\(\cos(A) < 1\)

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

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

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

### Problem 6

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

- What is the length of arc \(PQ\)?
- Explain why the length of arc \(PQ\) is less than \(\frac{1}{6}\) of the full circle.

### Problem 7

Label these points on the unit circle:

- \(Q\) is the image of \(P\) after a \(\frac{11\pi}{6}\) rotation with center \(O\).
- \(R\) is the image of \(P\) after a \(\frac{3\pi}{2}\) rotation with center \(O\).
- \(U\) is the image of \(P\) after a \(\frac{2\pi}{3}\) rotation with center \(O\).
- \(V\) is the image of \(P\) after a \(\frac{\pi}{3}\) rotation with center \(O\).