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 \((\cos(\frac{\pi}{3}),\sin(\frac{\pi}{3}))\)?

A:
A circle with center at the origin of an x y plane. Point A lies on the outside of the circle, in the first quadrant, and is closer to the x axis than the y axis.
B:
A circle with center at the origin of an x y plane.
C:
A circle with center at the origin of an x y plane.
D:
A circle with center at the origin of an x y plane

Problem 2

For which angles is the cosine positive? Select all that apply.

A:

0 radians

B:

\(\frac{5\pi}{12}\) radians

C:

\(\frac{5\pi}{6}\) radians

D:

\(\frac{3\pi}{4}\) radians

E:

\(\frac{5\pi}{3}\) 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?

Circle on a coordinate plane, center at the origin, radius 10 tick marks, no units given.

Problem 4

  1. For which angle measures, \(\theta\), between 0 and \(2\pi\) radians is \(\cos(\theta) = 0\)? Label the corresponding points on the unit circle.
    A circle with center at the origin of an x y plane on a grid. 
  2. 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 \((0.71,0.71)\).

A circle on a coordinate plane, center at the origin, B, radius 1. Points on the circle, A, at 1 comma 0, C in the first quadrant, D in the second quadrant, E in the fourth quadrant. Segment B C.
  1. The measure of angle \(ABD\) is \(\frac{3\pi}{4}\) radians. What are the approximate coordinates of \(D\)? Explain how you know.
  2. The measure of angle \(ABE\) is \(\frac{7\pi}{4}\) radians. What are the approximate coordinates of \(E\)? Explain how you know.
(From Unit 6, Lesson 4.)

Problem 6

  1. 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.
  2. Name 3 angles in this quadrant.
(From Unit 6, Lesson 4.)

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.  

(From Unit 5, Lesson 3.)