What is the equation of the tangent line of #r=cos^2(theta-pi) +sin^2(theta-3pi)-theta# at #theta=(-13pi)/4#?
1 Answer
#~~ (1.196)x + 17.406#
Explanation:
First, simplify
#r = cos^2(theta-pi) + sin^2(theta-3pi) - theta#
#= cos^2(theta-pi) + sin^2(theta-pi) - theta#
#= 1 - theta#
When
In cartesian coordinates, that would be
#x = rcostheta#
#= (1+{13pi}/4) * (-1/sqrt2)#
#~~ -7.927#
#y = rsintheta#
#= (1+{13pi}/4) * (1/sqrt2)#
#~~ 7.927#
The tangent line has to pass through the point
The gradient of the tangent line is equal to
#frac{"d"y}{"d"x} = frac{frac{"d"y}{"d" theta}}{frac{"d"x}{"d" theta}}#
#= frac{frac{"d"}{"d" theta}(rsintheta)}{frac{"d"}{"d" theta}(rcostheta)}#
#= frac{sinthetafrac{"d"}{"d" theta}(r)+rfrac{"d"}{"d" theta}(sintheta)}{costhetafrac{"d"}{"d" theta}(r)+rfrac{"d"}{"d" theta}(costheta)}#
#= frac{sinthetafrac{"d"r}{"d" theta}+rcostheta}{costhetafrac{"d"r}{"d" theta}-rsintheta}#
Next, we find
#frac{"d"r}{"d"theta} = frac{"d"}{"d" theta}(1-theta)#
#= -1#
Therefore,
#frac{"d"y}{"d"x} = frac{sintheta(-1)+(1 - theta)costheta}{costheta(-1)-(1 - theta)sintheta}#
#= frac{-sintheta+costheta-thetacostheta}{-costheta-sintheta+thetasintheta}# .
So, now we can find
#frac{"d"y}{"d"x}|_{theta = -{13pi}/4} #
#= frac{-sin(-{13pi}/4)+cos(-{13pi}/4)-(-{13pi}/4)cos(-{13pi}/4)}{-cos(-{13pi}/4)-sin(-{13pi}/4)+(-{13pi}/4)sin(-{13pi}/4)}#
#= frac{sin({13pi}/4)+cos({13pi}/4)+({13pi}/4)cos({13pi}/4)}{-cos({13pi}/4)+sin({13pi}/4)+({13pi}/4)sin({13pi}/4)}#
#= frac{(-1/sqrt2)+(-1/sqrt2)+({13pi}/4)(-1/sqrt2)}{-(-1/sqrt2)+(-1/sqrt2)+({13pi}/4)(-1/sqrt2)}#
#= frac{1+1+({13pi}/4)(1)}{-(1)+(1)+({13pi}/4)(1)}#
#= frac{2+{13pi}/4}{{13pi}/4}#
#= frac{8}{13pi}+1#
#~~= 1.196#
So now that we have the gradient of the line and one point that it passes through, we can write the equation of the line in point-slope form.
#y - ({4+13pi}/{4sqrt2}) = frac{8+13pi}{13pi}(x - (-{4+13pi}/{4sqrt2}))#
Now we rearrange the terms to get the slope-intercept form.
#y = frac{8+13pi}{13pi}x + (frac{8+13pi}{13pi}+1) {4+13pi}/{4sqrt2}#
#= frac{8+13pi}{13pi}x + frac{(4+13pi)^2}{26sqrt2pi}#
#~~ (1.196)x + 17.406#
Below is a graph for additional reference
