Question

What are the components of the vector between the origin and the polar coordinate `(3, (-7pi)/12)`?

Answer 1

They are `-3cos((5pi)/12)` horizontally and `3sin((5pi)/12)` vertically.

Explanation:

`(7pi)/12` is an angle in the second quadrant as shown here:
The angle AOB between the vector and the horizontal direction is `pi-(7pi)/12=(5pi)/12`
Dropping a vertical line down to the horizontal axis allows us to calculate the horizontal and vertical components of the vector with modulus 3 and argument `(7pi)/12`
The horizontal component is `3cos((5pi)/12)` and the vertical is `3sin((5pi)/12)`.
Since `(7pi)/12` is in the second quadrant, its `sin` is positive and its `cos` is negative, so the final components are:
`-3cos((5pi)/12)` horizontally and `3sin((5pi)/12)` vertically.