How would you solve for x using the double angle formula?

[bsmithysmith]

\(\displaystyle 2sin(2x)-3sin(x)=0\)

We did this in my class, but there were some parts where I was really confused. I know that we need to use the double angle formula, and the double angle formula for Sine is:

\(\displaystyle sin(2x) = 2sin(x)cos(x)\)

and correct me if I'm wrong. So what I had down is:

\(\displaystyle 2(2sin(x)cos(x))-3sin(x)=0\)

From thereon, I don't know where I should distribute it or something else. I checked Wolframalpha (just in case), and it showed:

\(\displaystyle 4cos(x)sin(x)-3sin(x)=0\)

I would think that the 2 would just distribute evenly, but how would I properly get through this part?

 

[MarkFL]

You are proceeding correctly. We are given:

\(\displaystyle 2\sin(2x)-3\sin(x)=0\)

Applying the double-angle identity for sine, we obtain:

\(\displaystyle 2\left(2\sin(x)\cos(x)\right)-3\sin(x)=0\)

\(\displaystyle 4\sin(x)\cos(x)-3\sin(x)=0\)

Now what you want to do is factor, and use the zero-factor property...

 

[bsmithysmith]

You are proceeding correctly. We are given:

\(\displaystyle 2\sin(2x)-3\sin(x)=0\)

Applying the double-angle identity for sine, we obtain:

\(\displaystyle 2\left(2\sin(x)\cos(x)\right)-3\sin(x)=0\)

\(\displaystyle 4\sin(x)\cos(x)-3\sin(x)=0\)

Now what you want to do is factor, and use the zero-factor property...

So I was correct after all (outside the forum)!

\(\displaystyle 4\sin(x)\cos(x)-3\sin(x)=0\)
\(\displaystyle sin(x)(4cos(x)-3)=0\)

so it's \(\displaystyle sin(x)=0\)
and \(\displaystyle cos(x)=3/4\)

For sine, \(\displaystyle x=0, pi\) and for cosine, it's \(\displaystyle 2pi-cos^1(3/4)\) and \(\displaystyle cos^-1(3/4)\) Note, it's Cosine inverse

 

[MarkFL]

Yes, those are the solutions if you are restricted to:

\(\displaystyle 0\le x<2\pi\)

To denote special characters using $\LaTeX$ like \(\displaystyle \pi\) and functions with fractions like \(\displaystyle \cos^{-1}\left(\frac{3}{4}\right)\), precede them with backslashes and use the frac command like so:

[noparsetex]\(\displaystyle \pi\)[/noparsetex]

[noparsetex]\(\displaystyle \cos^{-1}\left(\frac{3}{4}\right)\)[/noparsetex]

 

[CellCoree]

I would approach this problem by first understanding the concept of the double angle formula and how it relates to the given equation. The double angle formula for sine states that sin(2x) = 2sin(x)cos(x), which means that we can rewrite the given equation as:

2(2sin(x)cos(x)) - 3sin(x) = 0

Next, I would distribute the 2 to each term within the parentheses, resulting in:

4sin(x)cos(x) - 3sin(x) = 0

At this point, I would notice that both terms have a common factor of sin(x), so I would factor it out to simplify the equation:

sin(x)(4cos(x) - 3) = 0

Using the zero product property, we know that one of these factors must equal 0 in order for the entire equation to be true. Therefore, we can set each factor equal to 0 and solve for x:

sin(x) = 0 or 4cos(x) - 3 = 0

Solving for the first factor, we get x = 0 or x = π (since sin(0) = 0 and sin(π) = 0). For the second factor, we can use the inverse cosine function to solve for x:

cos(x) = 3/4

x = cos^-1(3/4) ≈ 0.7227 or x = 2π - cos^-1(3/4) ≈ 5.5606

Therefore, the solutions for x are x = 0, π, 0.7227, and 5.5606. These values can be verified by plugging them back into the original equation.

In summary, using the double angle formula and understanding the properties of trigonometric functions allowed us to solve for x in the given equation. As a scientist, it is important to have a strong understanding of mathematical concepts and the ability to apply them to solve problems.