Find the length in the first quadrant of the circle desribed by the polar equation r = 2sin theta + 4 cosTheta. The circle is graphed in the accompanying figure.
A. (Square Root 2)Pi
B. (Square Root 5)Pi
C. 2Pi
D. 5Pi
To find the length in the first quadrant of the circle described by the polar equation, we need to find the value of theta at the point where the circle intersects the x-axis.
Given the polar equation r = 2sin(theta) + 4cos(theta), we can rewrite it as r = 2sin(theta) + 4cos(theta) * (sin^2(theta) + cos^2(theta))^0.5.
Now, we convert the polar equation to Cartesian coordinates using the conversion formulas: x = rcos(theta) and y = rsin(theta).
x = (2sin(theta) + 4cos(theta))cos(theta) = 2sin(theta)cos(theta) + 4cos^2(theta)
y = (2sin(theta) + 4cos(theta))sin(theta) = 2sin^2(theta) + 4cos(theta)sin(theta)
Next, we set y = 0 since we are looking for the intersection with the x-axis.
2sin^2(theta) + 4cos(theta)sin(theta) = 0
We can factor out sin(theta) from the left side:
sin(theta)(2sin(theta) + 4cos(theta)) = 0
So either sin(theta) = 0 or (2sin(theta) + 4cos(theta)) = 0.
When sin(theta) = 0, theta can be 0 or pi.
When 2sin(theta) + 4cos(theta) = 0, we can rewrite this equation as sin(theta) = -2cos(theta).
Now, we divide both sides by cos(theta):
tan(theta) = -2
To find the values of theta where tan(theta) = -2 in the first quadrant, we take the inverse tangent (arctan) of -2.
theta = arctan(-2)
Using a calculator, we find the value of arctan(-2) to be approximately -63.4 degrees or -1.1 radians. However, we are interested in the value of theta in the first quadrant, which is between 0 and pi/2.
Since arctan(-2) falls in the fourth quadrant, we add pi radians to it to get the positive equivalent:
theta = arctan(-2) + pi
Using a calculator, we find arctan(-2) + pi to be approximately 2.14 radians.
Therefore, the length in the first quadrant of the circle is the value of theta where the circle intersects the x-axis, which is 2.14 radians.
Looking at the answer choices, the closest value to 2.14 radians is (A) √2π.