Find all possible values of θ for tan θ = 2 sin θ with the range 0◦ ≤ θ ≤ 360◦

First you rearrange the equation.It is known that tanθ=sinθ/cosθ therefore if we replace tanθ with sin/cos we get:sinθ/cosθ=2sinθIf you multiply both sides by cosθ this becomes:sinθ=2sinθcosθThen if you minus the sinθ from both sides you get: 0=2sinθcosθ-sinθAs sinθ is a common factor of both terms you can factorise it out 0=sinθ(2cosθ-1). For this statement to be true then either sinθ or 2cosθ-1 must equal to 0Therefore, we can use this to find values for θ.If sinθ=0 then arcsin(0)=θ and therefore θ= 0 (using a calculator) However, when you sketch the sin graph you can see that multiple values θ give a y value of 0 within the range. Reading from the graph you can see that 0,180 and 360 all give a value of 0. Therefore, θ could be all three values.We also need to solve 2cosθ-1 = 0. If we rearrange it we get cosθ=1/2 and then arccos(1/2)=60 (using a calculator). Like above, if we sketch the cos graph we can see that another value of θ gives the exact same y value as θ=60, and that the two values of θ are related by symmetry. Reading of the graph, we find this value to be 300°. This means that if we input 300 into 2cosθ-1 then we will also get 0.  Therefore, our possible values of θ are 0,180,360,60,300.