How do I solve 3sinx+2tanx=0 for 0≤x≤360 ?

The first step is to determine the domain of the functions involved. With the restriction  0≤x≤360 (degrees, I assume) we have exclude the points where  tan ( x )  is not defined.

By definition,  

tan ( x ) =

Therefore, we have to exclude points where  

cos ( x ) = 0 , that is the domain is described as

x ≠ 90  and  x ≠ 270

With the above restrictions in mind we can transform the equation as follows:

3 sin ( x ) + 2 sin( x ) /cos ( x ) = 0

The immediate temptation to reduce the above equation by  sin ( x )  should be accompanied by checking if it can be equal to zero to avoid losing solutions.

Indeed,  sin ( x ) = 0  can occur within a domain defined above.

It happens when

x = 0 ,  x =180  and  x =360 .

So the three values above are solutions.

After specifying these three solutions we can reduce the equation by  sin ( x )  getting

from which follows

solutions of this are

hope it will help :)

10 * 563.67 = 5636.7