# Lesson 5

The Pythagorean Identity (Part 1)

### Problem 1

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

### Solution

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### Problem 2

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

A:

B:

$$\frac{5\pi}{12}$$ radians

C:

$$\frac{5\pi}{6}$$ radians

D:

$$\frac{3\pi}{4}$$ radians

E:

$$\frac{5\pi}{3}$$ radians

### Solution

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### 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?

### Solution

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### Problem 4

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

### Solution

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### Problem 5

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

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

### Solution

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(From Unit 6, Lesson 4.)

### Problem 6

1. 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.
2. Name 3 angles in this quadrant.

### Solution

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(From Unit 6, Lesson 4.)

### 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.

### Solution

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(From Unit 5, Lesson 3.)