Convert from polar to rectangular r=3cos2theta
x= r cosTheta=3cos(2Theta)cosTheta
y= r sin Theta=3cos(2Theta)sinTheta
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r=3cos2theta
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Advanced Math: precal, trig identities, conversion
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Identities r everywhere!!! :/, thank you 4 the info appreciated
To convert from polar coordinates to rectangular coordinates, we need to use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
In this case, we are given the polar equation r = 3cos(2theta).
To convert this equation to rectangular coordinates, we substitute the given expression for r into the formulas for x and y:
x = (3cos(2theta)) * cos(theta)
y = (3cos(2theta)) * sin(theta)
Now, let's simplify these equations:
x = 3cos(2theta) * cos(theta)
= 3cos(2theta)cos(theta)
To simplify further, we can use the double angle identity for cosine:
cos(2theta) = cos^2(theta) - sin^2(theta)
Substituting this identity into the equation for x, we get:
x = 3(cos^2(theta) - sin^2(theta)) * cos(theta)
= 3(cos^3(theta) - sin^2(theta)cos(theta))
Similarly, for y:
y = 3cos(2theta) * sin(theta)
= 3(cos^2(theta) - sin^2(theta)) * sin(theta)
= 3(cos^2(theta)sin(theta) - sin^3(theta))
Therefore, the rectangular coordinates x and y of the polar equation r = 3cos(2theta) are:
x = 3(cos^3(theta) - sin^2(theta)cos(theta))
y = 3(cos^2(theta)sin(theta) - sin^3(theta))