Lesson 6

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

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

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.

Evaluate each of the following:

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

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

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 \(2(\sin(A))^2 = 1\).

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 \(\sin(A) = \sin(D)\).

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

\(\sin(\theta) < 0\)

\(\sin(\theta) > 0\)

\(\cos(\theta) < 0\)

\(\cos(\theta) > 0\)

\(\sin(\theta) > \cos(\theta)\)

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