Trig Quick Question: Solving for Values of θ in 0 ≤ θ ≤ 360 Interval

[phospho]

Solve for values of θ in the interval 0 ≤ θ ≤ 360,

2sinθ = cosecθ

now if I do:

2sinθ = 1/(sinθ) and multiply by sinθ

I can solve sinθ = ±√(1/2)

but if I solve 2sinθ - 1/(sinθ) = 0 and factorise to get sinθ(2 - (1/sin^2θ)) = 0

I get sinθ = 0 which gives me more solutions then needed. I've drawn the graphs of both functions and they meet where the solutions of θ are such that sinθ = ±√(1/2) , then why do I get more solutions if I factorise?

 

[Mentallic]

Because at $\sin\theta=0$ the other factor is undefined.

Think about the function $$y=\frac{x(x+1)}{x}$$ this is pretty much equivalent to $y=x+1$ except for the fact that it has a hole at x=0 (the coordinate (0,1)). If we solved this equation for when y=0, clearly the only answer is x=-1, but if we factored out x then what we'd have is:

$$x\left(\frac{x+1}{x}\right)=0$$

Which seems like it would have an x=0 solution, but it doesn't by the same reasoning.

 

[phospho]

Because at $\sin\theta=0$ the other factor is undefined.

Think about the function $$y=\frac{x(x+1)}{x}$$ this is pretty much equivalent to $y=x+1$ except for the fact that it has a hole at x=0 (the coordinate (0,1)). If we solved this equation for when y=0, clearly the only answer is x=-1, but if we factored out x then what we'd have is:

$$x\left(\frac{x+1}{x}\right)=0$$

Which seems like it would have an x=0 solution, but it doesn't by the same reasoning.

I see, thanks.

But why does this happen? Is there a particular reason or just one of those things.

 

[Mentallic]

I see, thanks.

But why does this happen? Is there a particular reason or just one of those things.

It happens because when we use the rule that if $ab=0$ then either a, b or both are equal to zero, we are assuming that a and b are real numbers. Undefined numbers and infinite are not real.

Without using any rigor, if we use an undefined number like 1/0 such that a=0 and b=1/0, then ab=1 (again, I'm abusing the maths here just to explain a point, don't take it as being correct). As you can see while we solved for a=0, it turns out that ab didn't turn out to be equal to zero, hence we cannot take a=0 because it is not a solution.

 

[phospho]

thank you