# Lesson 6

The Pythagorean Identity (Part 2)

Let’s use the Pythagorean Identity.

### 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.

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

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

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

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

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

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

### Problem 3

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

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

### Problem 4

Evaluate each of the following:

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

### 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.

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

### Problem 7

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

- 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 \(EF\).
- Explain why \(\sin(A) = \sin(D)\).

### Problem 8

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

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

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

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

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

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

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