Q: If $$ x = \cos(\theta) $$ and $$ y = \sin(\theta) $$, what does the point $$ (x, y) $$ represent on the unit circle?

A point on the unit circle corresponding to angle $$ \theta $$

Question 1

Which of the following equations represents the unit circle centered at the origin?

Accepted Answer

$$ x^2$ + y^2$ = 1 $$

Question 2

A point $$ (\frac{1}{2}, \frac{\sqrt{3}}{2})  is $$plotted on the coordinate plane. Is this point on the unit circle?

Accepted Answer

Yes, because $$ (\frac{1}{2})^2$ + (\frac{\sqrt{3}}{2})^2$ = 1 $$

Question 3

Which quadrant does the angle $$ \frac{5\pi}{6} $$ radians lie in?

Accepted Answer

Quadrant II

Question 4

Given the point $$ (0, -1) $$, is it on the unit circle?

Accepted Answer

Yes, because $$ 0^2$ + (-1)^2$ = 1 $$

Question 5

Which of the following points is NOT on the unit circle?

Accepted Answer

$$ (1, 1) $$

Question 6

If a point $$ (x, y)  is on $$the unit circle, which of the following must be true?

Accepted Answer

$$ x^2$ + y^2$ = 1 $$

Question 7

Which angle in radians corresponds to a point in Quadrant III on the unit circle?

Accepted Answer

$$ \frac{5\pi}{4} $$

Question 8

A point $$ (x, y) $$ satisfies $$ x^2$ + y^2$ = 1 . $$Which of the following could be the coordinates of this point?

Accepted Answer

$$ (0, -1) $$

Question 9

Which of the following best describes the unit circle?

Accepted Answer

A circle with radius 1 centered at the origin

Question 10

Test whether the point $$ (\frac{-\sqrt{2}}{2}, \frac{-\sqrt{2}}{2})  is on $$the unit circle.

Accepted Answer

Yes, because $$ (\frac{-\sqrt{2}}{2})^2$ + (\frac{-\sqrt{2}}{2})^2$ = 1 $$

Question 11

Which of the following is the general equation for a circle of radius $$ r $$ centered at $$ (h, k) $$?

Accepted Answer

$$ (x - h)^2$ + (y - k)^2$ = r^2$ $$

Question 12

If $$ x = \cos(	heta) $$ and $$ y = \sin(	heta) $$, what does the point $$ (x, y) $$ represent on the unit circle?

Accepted Answer

A point on the unit circle corresponding to angle $$ 	heta $$

Question 13

Which of the following angles is coterminal with $$ \frac{\pi}{3} $$?

Accepted Answer

$$ \frac{7\pi}{3} $$

Question 14

A real-world application: A Ferris wheel has a radius of 1 unit and is centered at the origin. If a seat is at the point $$ (0.6, 0.8) $$, is it on the edge of the Ferris wheel?

Accepted Answer

Yes, because $$ 0.6^2$ + 0.8^2$ = 1 $$

Question 15

Which of the following points is in Quadrant IV of the unit circle?

Accepted Answer

$$ (\frac{\sqrt{3}}{2}, -\frac{1}{2}) $$

Defining the Unit Circle: Introduction and Applications

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