Answered

Convert the point \( P=(r, \theta) \) to a rectangular coordinate of the form \((x, y)\):

A. \((x, y) = (\sqrt{2}, \sqrt{2})\)

B. \((x, y) = (2\sqrt{2}, 2\sqrt{2})\)

C. \((x, y) = \left(\frac{1}{2}, \frac{1}{2}\right)\)

D. [tex]\((x, y) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\)[/tex]



Answer :

GinnyAnswer
To convert the point \( P = (r, \theta) \) given in polar coordinates to rectangular coordinates \((x, y)\), we use the following formulas:
[tex]\[ x = r \cos(\theta) \][/tex]
[tex]\[ y = r \sin(\theta) \][/tex]

In this problem, we need to match the given list of rectangular coordinates to a point \( P = (r, \theta) \). According to the query, the point P has a radius \( r = 1 \) and an angle \( \theta = \frac{\pi}{4} \) radians (which is 45 degrees).

Following the steps:

1. First, we calculate the \( x \)-coordinate:
[tex]\[ x = 1 \cdot \cos\left(\frac{\pi}{4}\right) \][/tex]
[tex]\[ x = 1 \cdot \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ x = \frac{\sqrt{2}}{2} \][/tex]

2. Next, we calculate the \( y \)-coordinate:
[tex]\[ y = 1 \cdot \sin\left(\frac{\pi}{4}\right) \][/tex]
[tex]\[ y = 1 \cdot \frac{\sqrt{2}}{2} \][/tex]
[tex]\[ y = \frac{\sqrt{2}}{2} \][/tex]

Therefore, the rectangular coordinates for the point \( P = (r, \theta) \) are:
[tex]\[ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \][/tex]

Matching these coordinates with the given choices, we can see that the correct answer is:
[tex]\[ (x, y) = \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \][/tex]

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