What is the slope of the polar curve #f(theta) = sectheta - csctheta # at #theta = (3pi)/8#?
1 Answer
Explanation:
The slope of the tangent line of a function at a point is equal to the derivative of the function at that point.
With that being said, let's take the derivative
#(df)/(d theta) [f(theta) = sectheta - csctheta]#
The derivative of
#f'(theta) = tanthetasectheta - d/(d theta) [csctheta]#
The derivative of
#ul(f'(theta) = tanthetasectheta + cotthetacsctheta#
Or, in terms of
#ul(f'(theta) = (sintheta)/(cos^2theta) + (costheta)/(sin^2theta)#
Now, to find the slope at the point
#m = (sin((3pi)/8))/(cos^2((3pi)/8)) + (cos((3pi)/8))/(sin^2((3pi)/8)) = color(blue)(2sqrt(10+sqrt2)#
#= color(blue)(ulbar(|stackrel(" ")(" "6.757" ")|)#
