Question

What is the Cartesian form of `(45,(-13pi)/8)`?

Answer 1

`(45 cos(-(13pi)/8), 45 sin(-(13pi)/8))=(45/2sqrt(2-sqrt(2)), 45/2sqrt(2+sqrt(2)))`

Explanation:

The cartesian form of `(r, theta)` is `(rcos theta, rsin theta)`

Let's find `cos(-(13pi)/8)` and `sin(-(13pi)/8)` first.

Note that `-(13pi)/4` is coterminal with `(3pi)/4` in Q2.

Hence:

`{ (cos(-(13pi)/4) = -sqrt(2)/2), (sin(-(13pi)/4) = sqrt(2)/2) :}`

Then note that:

`cos 2theta = 2cos^2 theta - 1 = 1 - 2sin^2 theta`

Hence:

`cos(-(13pi)/8) = +-sqrt((1+cos(-(13pi)/4))/2)`

`color(white)(cos(-(13pi)/8)) = +-1/2sqrt(2+2cos(-(13pi)/4))`

`color(white)(cos(-(13pi)/8)) = +-1/2sqrt(2-sqrt(2))`

`sin(-(13pi)/8) = +-sqrt((1-cos(-(13pi)/4))/2)`

`color(white)(sin(-(13pi)/8)) = +-1/2sqrt(2-2cos(-(13pi)/4))`

`color(white)(sin(-(13pi)/8)) = +-1/2sqrt(2+sqrt(2))`

Which signs are correct?

Note that `-(13pi)/8` is coterminal with `(3pi)/8` which is in Q1.

So `cos` and `sin` are both positive and:

`{ (cos(-(13pi)/8) = 1/2sqrt(2-sqrt(2))), (sin(-(13pi)/8) = 1/2sqrt(2+sqrt(2))) :}`

So polar coordinates `(45, -(13pi)/8)` in rectangular coordinates is:

`(45 cos(-(13pi)/8), 45 sin(-(13pi)/8))=(45/2sqrt(2-sqrt(2)), 45/2sqrt(2+sqrt(2)))`