Question

What are the components of the vector between the origin and the polar coordinate `(12, (-3pi)/4)`?

Answer 1

`((-6sqrt2),(-6sqrt2))`

Explanation:

Using the formulae that links Polar to Cartesian coordinates,

`• x = rcostheta`

`• y = rsintheta`

Measured clock-wise from the x-axis the angle `(-(3pi)/4)`
places the point in the 3rd quadrant , where both the sine and cosine ratios are negative.
The related acute angle to `(3pi)/4 " is " pi/4`

and so `cos(-(3pi)/4) = -cos(pi/4)`
This is also the case for the sine ratio.

Using 'exact values' for these angles gives.

`x = -12cos(pi/4) = -12xx1/sqrt2`
and'rationalising' the denominator

`x= -12xx sqrt2/2 = -6sqrt2`

`y = -12sin(pi/4) = -6sqrt2`