# Lesson 6

The Pythagorean Identity (Part 2)

### Problem 1

The picture shows angles $$A$$ and $$B$$. Explain why $$\sin(B) = \text- \sin(A)$$ and why $$\cos(B) = \text-\cos(A)$$.

### Problem 2

Which statements are true? Select all that apply.

A:

$$\sin(\theta) > 0$$ for an angle $$\theta$$ in quadrant 2

B:

$$\cos(\theta) > 0$$ for an angle $$\theta$$ in quadrant 2

C:

$$\tan(\theta) > 0$$ for an angle $$\theta$$ in quadrant 2

D:

$$\sin(\theta) > 0$$ for an angle $$\theta$$ in quadrant 3

E:

$$\cos(\theta) > 0$$ for an angle $$\theta$$ in quadrant 3

F:

$$\tan(\theta) > 0$$ for an angle $$\theta$$ in quadrant 3

### Problem 3

The tangent of an angle satisfies $$\tan(\theta) = 10$$.

1. Which quadrant could $$\theta$$ lie in? Explain how you know.
2. Estimate the possible value(s) of $$\theta$$. Explain your reasoning.

### Problem 4

Evaluate each of the following:

1. $$\tan\left(\frac{5\pi}{4}\right)$$
2. $$\sin\left(\frac{3\pi}{2}\right)$$
3. $$\cos\left(\frac{7\pi}{4}\right)$$

### Problem 5

The sine of an angle $$\theta$$ in the second quadrant is $$0.6$$. What is $$\tan(\theta)$$? Explain how you know.

### Problem 6

Triangle $$ABC$$ is an isosceles right triangle in the unit circle.

1. Explain why $$\sin(A) = \cos(A)$$.
2. Use the Pythagorean Theorem to explain why $$2(\sin(A))^2 = 1$$.

### Solution

(From Unit 6, Lesson 5.)

### Problem 7

Triangle $$DEF$$ is similar to triangle $$ABC$$. The scale factor going from $$\triangle DEF$$ to $$\triangle ABC$$ is 3.

1. Explain why the length of segment $$AB$$ is 3 times the length of segment $$DE$$ and the length of segment $$BC$$ is 3 times the length of segment $$EF$$.
2. Explain why $$\sin(A) = \sin(D)$$.

### Solution

(From Unit 6, Lesson 2.)

### Problem 8

Which of the following is true for angle $$\theta$$? Select all that apply.

A:

$$\sin(\theta) < 0$$

B:

$$\sin(\theta) > 0$$

C:

$$\cos(\theta) < 0$$

D:

$$\cos(\theta) > 0$$

E:

$$\sin(\theta) > \cos(\theta)$$

F:

$$\sin(\theta) < \cos(\theta)$$