# Lesson 5

The Pythagorean Identity (Part 1)

- Let’s learn more about cosine and sine.

### Problem 1

The pictures show points on a unit circle labeled A, B, C, and D. Which point is \((\cos(\frac{\pi}{3}),\sin(\frac{\pi}{3}))\)?

### Problem 2

For which angles is the cosine positive? Select **all** that apply.

0 radians

\(\frac{5\pi}{12}\) radians

\(\frac{5\pi}{6}\) radians

\(\frac{3\pi}{4}\) radians

\(\frac{5\pi}{3}\) radians

### Problem 3

Mark two angles on the unit circle whose measure \(\theta\) satisfies \(\sin(\theta) = \text-0.4\). How do you know your angles are correct?

### Problem 4

- For which angle measures, \(\theta\), between 0 and \(2\pi\) radians is \(\cos(\theta) = 0\)? Label the corresponding points on the unit circle.
- What are the values of \(\sin(x)\) for these angle measures?

### Problem 5

Angle \(ABC\) measures \(\frac{\pi}{4}\) radians, and the coordinates of \(C\) are about \((0.71,0.71)\).

- The measure of angle \(ABD\) is \(\frac{3\pi}{4}\) radians. What are the approximate coordinates of \(D\)? Explain how you know.
- The measure of angle \(ABE\) is \(\frac{7\pi}{4}\) radians. What are the approximate coordinates of \(E\)? Explain how you know.

### Problem 6

- In which quadrant is the value of the \(x\)-coordinate of a point on the unit circle always greater than the \(y\)-coordinate? Explain how you know.
- Name 3 angles in this quadrant.

### Problem 7

Lin is comparing the graph of two functions \(g\) and \(f\). The function \(g\) is given by \(g(x) = f(x-2)\). Lin thinks the graph of \(g\) will be the same as the graph of \(f\), translated to the left by 2. Do you agree with Lin? Explain your reasoning.