Unit Circle Questions in Trigonometry

This page presents multiple-choice questions on the unit circle in trigonometry. The questions focus on identifying points on the unit circle, determining quadrants, and understanding how rotations affect coordinates. Detailed answers and explanations are provided at the bottom of the page.

Questions

Question 1

Which of the following points is on the unit circle?

a) \( (-\sqrt{2}/2 , -\sqrt{2}/2) \)
b) \( (\sqrt{2}/3 , -\sqrt{2}/3) \)
c) \( (1/2 , 1/2) \)
d) \( (3/2 , 2/3) \)

Question 2

A point is in Quadrant III and lies on the unit circle. If its x-coordinate is \( -4/5 \), what is the y-coordinate?

a) \( 3/5 \)
b) \( -3/5 \)
c) \( -2/5 \)
d) \( 5/3 \)

Question 3

Find the point on the unit circle associated with the rotation \( -9\pi/2 \).

a) \( (0 , -1) \)
b) \( (0 , 1) \)
c) \( (1 , 0) \)
d) \( (-1 , 0) \)

Question 4

Find the point on the unit circle associated with the angle \( 5\pi/3 \).

a) \( (1/2 , 1/2) \)
b) \( (-\sqrt{3}/2 , 1/2) \)
c) \( (1/2 , -\sqrt{3}/2) \)
d) \( (-\sqrt{3}/2 , -1/2) \)

Question 5

If the point \( (a , b) \) is on the unit circle associated with the rotation \( t \), which of the following statements is not correct?

a) \( \sin(t) = b \)
b) \( \cos(t) = a \)
c) \( \sin(-t) = -b \)
d) \( \cos(-t) = -a \)

Question 6

If the point \( (a , b) \) is on the unit circle associated with the rotation \( t \), and the point \( (c , d) \) is on the unit circle associated with the rotation \( t + \pi \), which of the following is correct?

a) \( c = -a \) and \( d = -b \)
b) \( c = -a \) and \( d = b \)
c) \( c = a \) and \( d = b \)
d) \( c = a \) and \( d = -b \)

Question 7

If the point \( (a , b) \) is on the unit circle associated with the rotation \( t \), which point corresponds to the rotation \( t + \pi/2 \)?

a) \( (b , a) \)
b) \( (-b , a) \)
c) \( (-b , -a) \)
d) \( (-a , b) \)

Answers and Explanations

1. a)
   A point \( (x,y) \) is on the unit circle if \( x^2 + y^2 = 1 \). \[ (-\sqrt{2}/2)^2 + (-\sqrt{2}/2)^2 = 1/2 + 1/2 = 1 \] So option (a) lies on the unit circle.
2. b)
   On the unit circle, \( x^2 + y^2 = 1 \). With \( x = -4/5 \): \[ (-4/5)^2 + y^2 = 1 \Rightarrow y^2 = 9/25 \] Thus \( y = \pm 3/5 \). Since the point is in Quadrant III, \( y \) must be negative, so \( y = -3/5 \).
3. d)
   Reduce the angle: \[ -9\pi/2 = -4\pi - \pi/2 \] This is equivalent to \( -\pi/2 \), which corresponds to the point \( (-1,0) \).
4. c)
   The angle \( 5\pi/3 \) is in Quadrant IV. On the unit circle: \[ (\cos 5\pi/3, \sin 5\pi/3) = (1/2 , -\sqrt{3}/2) \] So option (c) is correct.
5. d)
   For any angle \( t \): \[ \sin(-t) = -\sin(t), \quad \cos(-t) = \cos(t) \] Thus \( \cos(-t) = -a \) is false, making option (d) incorrect.
6. a)
   Adding \( \pi \) to an angle moves the point to the opposite side of the unit circle: \[ (\cos(t+\pi), \sin(t+\pi)) = (-a, -b) \] So option (a) is correct.
7. b)
   A rotation of \( \pi/2 \) corresponds to a counterclockwise rotation by 90°: \[ (\cos(t+\pi/2), \sin(t+\pi/2)) = (-b, a) \] Hence option (b) is correct.

More References

Trigonometry Problems and Tutorials