Question

What are the angles that would satisfy the following trigonometric equation below, ranging from 0 degree to 360 degrees?

`6 cosx - 5sinx = 0`

Answer 1

The solutions are `x=arctan(6/5),quad180^@+arctan(6/5)`.

Explanation:

`6cosx-5sinx=0`

`6cosx=5sinx`

Divide by `cosx`:

`(6cosx)/cosx=(5sinx)/cosx`

`(6color(red)cancelcolor(black)cosx)/color(red)cancelcolor(black)cosx=(5sinx)/cosx`

`6=(5sinx)/cosx`

`6=5*sinx/cosx`

`6=5*tanx`

`6/5=(5*tanx)/5`

`6/5=(color(red)cancelcolor(black)5*tanx)/color(red)cancelcolor(black)5`

`6/5=tanx`

`tanx=6/5`

`x=arctan(6/5),quad180^@+arctan(6/5)`

`color(white)x~~50.1944^@,230.1944^@`

The reason there are two answers is that the `tan` function is periodic every `180^@`, so we can add `180^@` and the answer will still be between `0^@` and `360^@`.

Hope this helped!