Convert the polar equation \( r = 4\cos(\theta) \) to Cartesian coordinates. - 3.13 Trigonometry and Polar Coordinates - Review - Use this assignment in your class!

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3.13 Trigonometry and Polar Coordinates - Review

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Explain how you can convert the polar coordinates

(r, θ)

to Cartesian coordinates

(x, y)

Describe the process of converting Cartesian coordinates

(x, y)

to polar coordinates

(r, θ)

How can you determine the angle

θ

in polar coordinates if you are given Cartesian coordinates

(x, y)

and

x

is negative?

How can you use trigonometric functions to verify that a given set of polar coordinates

(r, θ)

correctly converts to Cartesian coordinates

(x, y)

Graph the polar equation

r = 2 + 2 \cos (θ)

and describe its shape.

Convert the polar coordinates

(4, \frac{π}{3})

to Cartesian coordinates.

Describe the relationship between the polar equation

r = 3 \sin (θ)

and its graph.

Explain how to plot the point given in polar coordinates

(5, \frac{2 π}{3})

on a polar graph.

Determine the polar coordinates of the point where a line with the equation

y = x

intersects the circle

x^{2} + y^{2} = 8

A Ferris wheel has a radius of 10 meters and completes one full rotation every 2 minutes. Write the parametric equations for the position of a point on the Ferris wheel in polar coordinates as a function of time

t

Convert the polar equation

r = 4 \cos (θ)

to Cartesian coordinates.

Convert the polar equation

r = 4 \cos (θ)

to Cartesian coordinates.

To convert

r = 4 \cos (θ)

to Cartesian coordinates, use the relationships

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

and

\cos (θ) = \frac{x}{r}

. Substituting these into the polar equation gives

\sqrt{x^{2} + y^{2}} = 4 \frac{x}{\sqrt{x^{2} + y^{2}}}

. Multiplying both sides by

\sqrt{x^{2} + y^{2}}

results in

x^{2} + y^{2} = 4 x

. Rearranging gives

x^{2} - 4 x + y^{2} = 0

. Completing the square for

x

results in

(x - 2)^{2} + y^{2} = 4

, which is the equation of a circle with center (2, 0) and radius 2.

Which of the following is the Cartesian form of the polar equation

r = 4 \cos (θ)

(x + 2)^{2} + y^{2} = 4

x^{2} + y^{2} = 4

(x - 2)^{2} + y^{2} = 4

x^{2} + y^{2} = 16

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