Find an equation (in terms of x and y) of the tangent line to the curve r=3sin(5theta) at theta=pi/3
r = 3sin5Į
since x = rcosĮ and y = rsinĮ
dy/dx = dy/dĮ / dx/dĮ
= (r'sinĮ + rcosĮ)/(r'cosĮ - rsinĮ)
= (15cos5ĮsinĮ + 3sin5ĮcosĮ)/(15cos5ĮcosĮ - 3sin5ĮsinĮ)
= (15 * 1/2 * �ã3/2 + 3 * (-�ã3/2) * 1/2)
--------------------------------
(15 * 1/2 * 1/2 - 3 * -�ã3/2 * �ã3/2)
= (15�ã3/4 - 3�ã3/4)/(15/4 + 9/4)
= 12�ã3 / 24
= �ã3/2
So, now you have a point (pi/3,-3�ã3/2) and a slope.
(y+�ã3/2) = �ã3/2 (x-pi/3)
Interpret
Į as theta
�ã as sqrt
sorry.
To find an equation of the tangent line to the curve at a given point, you need to find the slope of the tangent line and the coordinates of the point.
First, let's find the coordinates of the point on the curve at theta = pi/3. Substitute theta = pi/3 into the polar equation r = 3sin(5theta):
r = 3sin(5(pi/3))
r = 3sin(5/3 * pi)
To convert the polar coordinates (r, theta) to Cartesian coordinates (x, y), we can use the following formulas:
x = r * cos(theta)
y = r * sin(theta)
Substituting the value of r from above and theta = pi/3 into the Cartesian coordinate formulas:
x = 3sin(5/3 * pi) * cos(pi/3)
y = 3sin(5/3 * pi) * sin(pi/3)
Now, let's find the slope of the tangent line. The slope of a curve in polar coordinates is given by the derivative of r with respect to theta (dr/dθ), divided by the derivative of theta with respect to theta (dθ/dθ = 1):
m = (dr/dθ) / (dθ/dθ)
Differentiating the equation r = 3sin(5θ) with respect to θ, we get:
dr/dθ = 3 * d(sin(5θ))/dθ
dr/dθ = 15cos(5θ)
After finding the slope (m) at the given point, substitute the coordinates (x, y) and the slope (m) into the point-slope form of a linear equation:
y - y1 = m(x - x1),
where (x1, y1) are the coordinates of the point.
Let's calculate these values.
The coordinates of the point on the curve at theta = pi/3 are:
x = 3sin(5/3 * pi) * cos(pi/3)
y = 3sin(5/3 * pi) * sin(pi/3)
The derivative of r with respect to theta is:
dr/dθ = 15cos(5θ)
Now we can substitute the values into the point-slope form equation:
y - y1 = m(x - x1)
where x1 = 3sin(5/3 * pi) * cos(pi/3) and y1 = 3sin(5/3 * pi) * sin(pi/3).
This equation will give you the equation of the tangent line to the curve at theta = pi/3, in terms of x and y.