What is the Cartesian form of (-10,(14pi)/8))(−10,14π8))?

2 Answers
smendyka
Jan 14, 2018

See a solution process below:

Explanation:

The formula for converting Polar Coordinates to Cartesian coordinates is:

For, (r, theta)(r,θ); x = r xx cos(theta)x=r×cos(θ); y = r xx sin(theta)y=r×sin(θ)

Substituting the coordinates from the problem gives:

For, (-10, (14pi)/8)(−10,14π8):

x = -10 xx cos((14pi)/8) = -10 xx 0.707 = -70.7x=−10×cos(14π8)=−10×0.707=−70.7

y = -10 xx sin((14pi)/8) = -10 xx -0.707 = 70.7y=−10×sin(14π8)=−10×−0.707=70.7

(-10, (14pi)/8) =(-70.7, 70.7)(−10,14π8)=(−70.7,70.7)

turksvids
Jan 14, 2018

The cartesian form is (-5sqrt(2), 5sqrt(2))(−5√2,5√2).

Explanation:

To convert polar to rectangular we use:

x=rcos(theta)x=rcos(θ)
y=rsin(theta)y=rsin(θ)

We're given the polar point (-10,(14pi)/8)(−10,14π8).

First let's simplify (14pi)/814π8 to (7pi)/47π4.

From what we have we know:

x=(-10)cos((7pi)/4)x=(−10)cos(7π4)
y=(-10)sin((7pi)/4)y=(−10)sin(7π4)

(7pi)/47π4 is a Unit Circle angle so we know its sine and cosine.

cos((7pi)/4)=sqrt(2)/2cos(7π4)=√22
sin((7pi)/4)=-sqrt(2)/2sin(7π4)=−√22

so:

x=(-10)(sqrt(2)/2)=-5sqrt(2)x=(−10)(√22)=−5√2
y=(-10)(-sqrt(2)/2)=5sqrt(2)y=(−10)(−√22)=5√2

so the cartesian form is (-5sqrt(2), 5sqrt(2))(−5√2,5√2).

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