QuestionProb. 5 Consider these three diagrams. (a) The first diagram depicts a point lying on a circle of radius 1 centered at the point which is to say, the unit circle. What are the coordinates of this point? (b) The second diagram depicts a point lying on a circle of radius 2 centered at the point . (Effectively, we've made the previous circle twice as big.) What are the coordinates of this point? (c) The third diagram depicts a point lying on a circle of radius 2 centered at the point . (Effectively, we've shifted the previous circle 2 units left and 1 unit up.) What are the coordinates of this point?
Studdy Solution
STEP 1
1. The angle is measured from the positive x-axis in standard position.
2. The coordinates of a point on a circle can be determined using the circle's radius and the angle from the positive x-axis.
STEP 2
1. Determine the coordinates of on the unit circle.
2. Determine the coordinates of on the circle of radius 2 centered at the origin.
3. Determine the coordinates of on the circle of radius 2 centered at .
STEP 3
To find , use the unit circle formula. For a circle of radius 1, the coordinates are given by:
\[
p_1 = \cos\left(\frac{2\pi}{3}\right) $
\[
q_1 = \sin\left(\frac{2\pi}{3}\right) $
Calculate:
\[
p_1 = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} $
\[
q_1 = \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} $
Thus, .
STEP 4
To find , use the formula for a circle of radius 2:
\[
p_2 = 2 \cdot \cos\left(\frac{2\pi}{3}\right) $
\[
q_2 = 2 \cdot \sin\left(\frac{2\pi}{3}\right) $
Calculate:
\[
p_2 = 2 \cdot \left(-\frac{1}{2}\right) = -1 $
\[
q_2 = 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3} $
Thus, .
STEP 5
To find , use the transformation for a circle of radius 2 centered at :
\[
p_3 = -2 + 2 \cdot \cos\left(\frac{2\pi}{3}\right) $
\[
q_3 = 1 + 2 \cdot \sin\left(\frac{2\pi}{3}\right) $
Calculate:
\[
p_3 = -2 + 2 \cdot \left(-\frac{1}{2}\right) = -3 $
\[
q_3 = 1 + 2 \cdot \frac{\sqrt{3}}{2} = 1 + \sqrt{3} $
Thus, .
The coordinates for each point are:
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