Rewrite the following equation in polar form. x^2+(y-5)^2 = 25
Show all your work. Your final answer should be r=
Don't skip any steps and don't explain with words, show the steps of the equation.
x^2+(y-5)^2 = 25
x^2+y^2-10y+25 = 25
x^2+y^2 = 10y
r^2 = 10rsin(theta)
r = 10sin(theta)
e
The mathematical constant e is approximately equal to 2.71828. It is an irrational number, which means it cannot be expressed as a simple fraction. e is a fundamental constant in calculus and appears in many mathematical formulas, such as the formula for compound interest. It is also closely related to the natural logarithm function.
To rewrite the equation x^2 + (y - 5)^2 = 25 in polar form, we need to express the equation in terms of the polar coordinates (r, θ). Here are the steps to do so:
Step 1: Replace x with r * cos(θ) and y with r * sin(θ).
(r * cos(θ))^2 + (r * sin(θ) - 5)^2 = 25
Step 2: Expand the equation.
r^2 * cos^2(θ) + (r * sin(θ) - 5)^2 = 25
Step 3: Simplify and apply trigonometric identities.
r^2 * cos^2(θ) + r^2 * sin^2(θ) - 10r * sin(θ) + 25 = 25
Step 4: Combine like terms.
r^2 * (cos^2(θ) + sin^2(θ)) - 10r * sin(θ) + 25 = 25
Step 5: Apply the trigonometric identity cos^2(θ) + sin^2(θ) = 1.
r^2 - 10r * sin(θ) + 25 = 25
Step 6: Simplify.
r^2 - 10r * sin(θ) = 0
Finally, rearranging the equation gives us the polar form:
r = 10 * sin(θ)
Hence, the final answer is r = 10 * sin(θ).