Can you please help me with this question on my study guide for my trig final? Thanks! I'd also appreciate if you can check out my other postings and check my work for some other study guide questions. I want to make sure I'm ready for my final. :)
4. Transform each polar equation to an equation in rectangular coordinates and identify its shape.
a. r = 6
b. r = 2cosθ
Answer:
r=6 is just a circle of radius 6
x^2+y^2 = r^2
This is also a circle, but shifted along the x-axis:
r = 2cosθ
r^2 = 2rcosθ
x^2+y^2 = 2x
x^2-2x+1 + y^2 = 1
(x-1)^2 + y^2 = 1
Of course, I'd be happy to help you with your trigonometry questions! Let's go over each part of the question one by one.
a. To transform the polar equation r = 6 into rectangular coordinates, we can use the following conversions:
- r = √(x^2 + y^2)
- x = r * cosθ
- y = r * sinθ
Plugging in the given value r = 6 into the above equations, we have:
- √(x^2 + y^2) = 6
Simplifying this equation, we obtain:
- x^2 + y^2 = 36
So the polar equation r = 6 can be transformed into the rectangular equation x^2 + y^2 = 36. This equation represents a circle with a radius of 6 centered at the origin.
b. To transform the polar equation r = 2cosθ into rectangular coordinates, we can apply the same conversions as in part a.
Plugging in the given value r = 2cosθ into the equations, we have:
- √(x^2 + y^2) = 2cosθ
Squaring both sides of the equation gives:
- x^2 + y^2 = 4cos^2θ
Now, using the trig identity cos^2θ = 1 - sin^2θ, we can rewrite the equation as:
- x^2 + y^2 = 4(1 - sin^2θ)
Simplifying further, we have:
- x^2 + y^2 = 4 - 4sin^2θ
Since sin^2θ = (y/r)^2 = y^2/(x^2 + y^2), we can substitute this expression into the equation:
- x^2 + y^2 = 4 - 4(y^2/(x^2 + y^2))
Simplifying again, we obtain:
- x^2 + y^2 = 4 - 4y^2/(x^2 + y^2)
Thus, the polar equation r = 2cosθ can be transformed into the rectangular equation x^2 + y^2 = 4 - 4y^2/(x^2 + y^2). This equation represents a limaçon-shaped curve.
I hope this helps! Let me know if you have any more questions or if you'd like me to check your work on other study guide questions.