Lesson 5

The Pythagorean Identity (Part 1)

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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

Solution

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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

Solution

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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.

Solution

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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?

Solution

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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.

Solution

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(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.

Solution

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(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.  

Solution

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(From Unit 5, Lesson 3.)