Questions: A point is given in rectangular coordinates. Convert the point to polar coordinates. (There are many correct answers.) (r, θ)=((3,3))

Solution Steps

To find the radius \( r \) in polar coordinates, we use the formula:

\[ r = \sqrt{x^2 + y^2} \]

Substituting the given values \( x = 3 \) and \( y = 3 \):

\[ r = \sqrt{3^2 + 3^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

The angle \( \theta \) can be calculated using the arctangent function:

\[ \theta = \arctan\left(\frac{y}{x}\right) \]

Substituting the values:

\[ \theta = \arctan\left(\frac{3}{3}\right) = \arctan(1) \]

The angle \( \theta \) in radians is:

\[ \theta = \frac{\pi}{4} \]

To convert the angle \( \theta \) from radians to degrees, we use the conversion factor \( \frac{180}{\pi} \):

\[ \theta_{\text{degrees}} = \theta \cdot \frac{180}{\pi} = \frac{\pi}{4} \cdot \frac{180}{\pi} = 45^\circ \]

The polar coordinates \( (r, \theta) \) are then expressed as:

\[ (r, \theta) = \left(3\sqrt{2}, 45^\circ\right) \]

Final Answer