Question

How do you solve `2sinx+cscx=0` in the interval 0 to 2pi?

Answer 1

No solution.

Explanation:

`2sinx+cscx=0` is equivalent to

`2sinx+1/sinx=0` - assuming `sinx!=0` and multiplying by `sinx`

`2sin^2x+1=0` or `sin^2x=-1/2`

But as `sin^2x` cannot be negative, we do not have any solution to `2sinx+cscx=0`

If we draw the graph of `2sinx+cscx`, it is observed that it never touches `x`-axis and hence no solution.

graph{2sinx+cscx [-10, 10, -5, 5]}