Lesson 5
The Pythagorean Identity (Part 1)
- Let’s learn more about cosine and sine.
Problem 1
The pictures show points on a unit circle labeled A, B, C, and D. Which point is \((\cos(\frac{\pi}{3}),\sin(\frac{\pi}{3}))\)?
Problem 2
For which angles is the cosine positive? Select all that apply.
0 radians
\(\frac{5\pi}{12}\) radians
\(\frac{5\pi}{6}\) radians
\(\frac{3\pi}{4}\) radians
\(\frac{5\pi}{3}\) radians
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?
Problem 4
- For which angle measures, \(\theta\), between 0 and \(2\pi\) radians is \(\cos(\theta) = 0\)? Label the corresponding points on the unit circle.
- What are the values of \(\sin(x)\) for these angle measures?
Problem 5
Angle \(ABC\) measures \(\frac{\pi}{4}\) radians, and the coordinates of \(C\) are about \((0.71,0.71)\).
- The measure of angle \(ABD\) is \(\frac{3\pi}{4}\) radians. What are the approximate coordinates of \(D\)? Explain how you know.
- The measure of angle \(ABE\) is \(\frac{7\pi}{4}\) radians. What are the approximate coordinates of \(E\)? Explain how you know.
Problem 6
- 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.
- Name 3 angles in this quadrant.
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.