Math Snap

STEP 1

What is this asking?
We're looking for all the angles, $x$, that make this trigonometric equation true!
Watch out!
Don't divide both sides by $\sin(x)$!
You might lose some solutions!

STEP 2

1. Rewrite the equation
2. Consider the zero case
3. Consider the non-zero case

STEP 3

Alright, let's rewrite our equation to make it easier to work with!
We'll move everything to one side by subtracting $3\sin(x)$ from both sides.
This gives us:
$$2\sin(x)\cos(x) - 3\sin(x) = 0$$

STEP 4

Now, we can factor out $\sin(x)$, which is super helpful!
This gives us:
$$\sin(x) \cdot (2\cos(x) - 3) = 0$$

STEP 5

This equation is true if either $\sin(x) = 0$ or $2\cos(x) - 3 = 0$.
Let's tackle $\sin(x) = 0$ first.
When does $\sin(x)$ equal zero?

STEP 6

Well, $\sin(x) = 0$ when $x$ is a multiple of $\pi$!
So, $x = n\pi$, where $n$ can be any integer (like 0, 1, 2, -1, -2, and so on).
We've found some solutions already!

STEP 7

Now, let's explore the other possibility: $2\cos(x) - 3 = 0$.
First, we isolate $\cos(x)$ by adding 3 to both sides and then dividing by 2:
$$2\cos(x) = 3$$ $$\cos(x) = \frac{3}{2}$$

STEP 8

Hold on a second!
The cosine function can never be greater than one or less than negative one.
Since $\frac{3}{2}$ is greater than one, there are no solutions for this part of the equation!
That means we've already found all the solutions in the previous step!

The solutions to the equation $2\sin(x)\cos(x) = 3\sin(x)$ are $x = n\pi$, where $n$ is any integer.