What is the slope of the polar curve $f (θ) = sec θ - csc θ$ $f(theta) = sectheta - csctheta$ at $θ = \frac{5 π}{4}$ $theta = (5pi)/4$ ?

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Answer 1 · 2017-09-03T04:15:45

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Answer 2 · 2017-09-04T16:38:24

$1$ $1$

$r = f (θ) = sec θ - csc θ$ $r=f(theta)=sectheta-csctheta$

We want $\frac{d y}{d x}$ $dy/dx$ of this curve at $θ = \frac{5 π}{4}$ $theta=(5pi)/4$ .

To find the slope in terms of $x$ $x$ and $y$ $y$ , we have to use the identities ${(x=rcostheta),(y=rsintheta):}$

So here,

${(x=(sectheta-csctheta)costheta=1-cottheta),(y=(sectheta-csctheta)sintheta=tantheta-1):}$

Note that $dy/dx=(dy/(d theta))/(dx/(d theta))$ .

${(dx/(d theta)=d/(d theta)(1-cottheta)=csc^2theta),(dy/(d theta)=d/(d theta)(tantheta-1)=sec^2theta):}$

So:

$dy/dx=(dy/(d theta))/(dx/(d theta))=sec^2theta/csc^2theta=tan^2theta$

So the slope at $theta=(5pi)/4$ is:

$m=tan^2((5pi)/4)=(-1)^2=1$

What is the slope of the polar curve f(theta) = sectheta - csctheta f(θ)=secθ−cscθf(theta) = sectheta - csctheta at theta = (5pi)/4θ=5π4theta = (5pi)/4?