How do you solve this problem?
y = 2sin 3x + 4sin 2x
I haven't done these types of problems for awhile and have forgotten how to do them.
2 sin 3x = 2[ 3 sin x-4 sin^3 x]
= 6 sin x - 8 sin^3 x
4 sin 2x = 4[ 2 sin x cos x ]
= 8 sin x cos x
so
6 sin x - 8 sin^3 x + 8 sin x cos x
2 sin x [ 3 - 4 sin^2 x + 4 cos x ]
To solve the problem y = 2sin 3x + 4sin 2x, we'll need to apply the properties of trigonometric functions. Specifically, we'll use the double-angle and triple-angle identities.
Step 1: Simplify the equation using the double-angle and triple-angle identities.
To simplify the equation, we'll use the following identities:
- sin 2x = 2sin x cos x (double-angle identity)
- sin 3x = 3sin x - 4sin^3 x (triple-angle identity)
Substituting these identities into the original equation, we get:
y = 2(3sin x - 4sin^3 x) + 4(2sin x cos x)
Simplifying further, we have:
y = 6sin x - 8sin^3 x + 8sin x cos x
Step 2: Simplify the equation further, if possible.
Combining like terms, we can rewrite the equation as:
y = 14sin x - 8sin^3 x + 8sin x cos x
Step 3: Analyze the equation.
Now that we have simplified the equation, we can analyze it to see its properties and characteristics. For example, we can observe that y is composed of sine and cosine terms with various coefficients.
Step 4: Use specific values or techniques to solve.
Depending on the specific problem or what you're trying to achieve, you may need to solve for specific values or use additional techniques to determine the behavior of the equation. This could involve finding the amplitude, period, or points where the equation is equal to zero.
Remember, the solution to this problem was found by applying trigonometric identities and algebraic manipulation. The specific solution or interpretation will depend on the context and purpose of your problem.