craig16 said:Homework Statement
x=2acos(theta- pi/3)cos(theta)
y=2acos(theta- pi/3)sin(theta)
Write everything in terms of x and y
craig16 said:Yaaa, i figured it out.
Just multiplying both sides of the r function by r, then subbing in y=rsin(theta) and x=rcos(theta) then using sin^2 + cos^2 =1 will make it just in terms of x and y.
To convert a polar equation to a Cartesian equation, we can use the following trigonometric identities:
x = r cos θ
y = r sin θ
where r represents the distance from the origin and θ represents the angle from the positive x-axis to the point.
Using these identities, we can rewrite the polar equation in terms of x and y.
Yes, all polar equations can be rewritten using trigonometric identities. These identities help us to convert the polar coordinates (r, θ) into Cartesian coordinates (x, y).
Rewriting a polar equation as a Cartesian equation can be useful in graphing and analyzing the equation. It allows us to plot the equation on a rectangular coordinate system and easily identify important points such as the x and y-intercepts, symmetry, and asymptotes.
The Pythagorean identity, sin² θ + cos² θ = 1, can be used to rewrite a polar equation as a Cartesian equation. For example, if the polar equation is r = 2 cos θ, we can square both sides to get r² = 4 cos² θ. Then, using the Pythagorean identity, we can replace cos² θ with 1 - sin² θ to get r² = 4(1 - sin² θ). Finally, we can use the identity x² + y² = r² to rewrite the equation in terms of x and y.
Yes, there are a few special cases to consider when rewriting a polar equation as a Cartesian equation. These include equations that involve the polar coordinates (r, θ) being raised to a power, equations with multiple angles, and equations with negative angles. In these cases, we may need to use additional trigonometric identities and algebraic manipulations to rewrite the equation in terms of x and y.