Can Someone Explain This Example in More Depth? (Trigonometric Equation Question)

[teetar]

I'm trying to figure out how this question was solved:
Solve for θ where 0 [FONT=MathJax_AMS]⩽[/FONT] θ [FONT=MathJax_AMS]⩽[/FONT] 2π:
2sin²θ = sinθ
Here is the given solution:
2sin²θ = sinθ
2sin²θ - sinθ = 0
sinθ(2sinθ - 1) = 0
sinθ = 0 or 1/2
θ = 0, π/6, 5π/6, π, or 2π

I just don't understand what is happening after the "sinθ(2sinθ - 1) = 0" step. I don't know where that 1 comes from, and I don't know what is happening after that at all. I understand this is "simple algebra," but I don't understand it at all so far, so I was just hoping someone might be able to do out this problem in more detail to fill me in on what I'm not seeing.

Any help will be greatly appreciated, and I'm sorry to have to ask a seemingly simple question like this one.
Thanks!

 

[Quaid]

Solve for θ where 0 [FONT=MathJax_AMS]⩽[/FONT] θ [FONT=MathJax_AMS]⩽[/FONT] 2π:

2sin²θ = sinθ

Here is the given solution:

2sin²θ = sinθ

2sin²θ - sinθ = 0

sinθ(2sinθ - 1) = 0

sinθ = 0 or 1/2

θ = 0, π/6, 5π/6, π, or 2π

don't understand what is happening after the "sinθ(2sinθ - 1) = 0" step. I don't know where that 1 comes from

I don't know what is happening after that at all.

Hi teetar:

The 1 comes from factoring. Here is a basic example of factoring that follows the same form.

2x^2 - x = 0

The left-hand side means 2*x*x - x

In other words, we have two terms, and each of these terms contains an x, so we can factor out an x.

(x)(2x - 1)

This factorization must multiply back out to equal the original 2x^2-x. The only way that can happen is if we subtract 1 from 2x, as shown.

Do you remember the Distributive Property? Well, factoring is kinda like the reverse of distributing.

We multiply out (x)(2x - 1) by distributing the x across the terms inside the parentheses.

x(2x) - x(1) = 2x^2 - x

The same factoring situation occurs in your exercise. The -1 needs to appear in the factorization so that we get back the original expression (if it's multiplied out). Otherwise, it would not be a correct factorization.

I hope the -1 is clear now, but let us know if you're still puzzling over it.

The next step is known as the Zero-Product Property. When the result of multiplying two factors is zero, this property tells us that one or both of those factors must be zero. (It makes sense, if you ponder it. How else could you get zero by multiplying, if you are not multiplying by zero? You can't!)

Therefore, whenever you're solving an equation of the form (stuff)*(other stuff) = 0, you set each factor equal to zero.

stuff = 0

other stuff = 0

Then, solve each equation. This is what they did.

(sinθ)(2sinθ-1) = 0

Solve sinθ = 0

Solve 2sinθ - 1 = 0

You know how to get sinθ=1/2, from the second equation above, yes?

Finally, the solutions to these two equations are special, known angles that beginning students need to memorize.

Look up "special angles" in your textbook's index; hopefully, your text has a chart of special angles. Let us know, if you can't find them.

I'm sorry to have to ask a seemingly simple question like this one.

Well, DON'T BE!!

There is no need for sorrow, when you don't understand something and have to ask a question. That's how we all learn math!

Cheers

 

[teetar]

Thanks!

Thanks a bunch for that, @Quaid, your help has made everything quite understandable for me! I can see now the steps I was previously missing, and for that I am very grateful.
Have a good one, man!