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.
\(\sin(\theta) < 0\)
\(\sin(\theta) > 0\)
\(\cos(\theta) < 0\)
\(\cos(\theta) > 0\)
\(\sin(\theta) > \cos(\theta)\)
\(\sin(\theta) < \cos(\theta)\)