sin(7x)= sin(x)[cos^2(3x)-sin^2(3x)]+2cos(x)cos(3x)sin(3x)
I tried for an hour but still don't know how. Plzz help
sin(7x) = sin(x+6x)
= sinx cos6x + cosx sin6x
= sinx*cos(2*3x) + cosx*sin(2*3x)
...
To solve this equation, we can simplify both sides first.
Let's start with the right-hand side (RHS) of the equation:
sin(x)[cos^2(3x) - sin^2(3x)] + 2cos(x)cos(3x)sin(3x)
We can use the trigonometric identity cos^2(A) - sin^2(A) = cos(2A) to simplify the first term:
sin(x) * [cos(2*3x)] + 2cos(x)cos(3x)sin(3x)
Now we can simplify further:
sin(x) * cos(6x) + 2cos(x)cos(3x)sin(3x)
Next, we can expand the double-angle identity for cos(6x):
sin(x) * [cos^2(3x) - sin^2(3x)] + 2cos(x)cos(3x)sin(3x)
Now, the equation becomes:
sin(x)cos^2(3x) - sin(x)sin^2(3x) + 2cos(x)cos(3x)sin(3x)
Distribute the sin(x) to both terms:
sin(x)cos^2(3x) - sin^3(3x) + 2cos(x)cos(3x)sin(3x)
Now we have simplified the right-hand side of the equation. Let's focus on the left-hand side (LHS) of the equation:
sin(7x)
At this point, the equation becomes:
sin(x)cos^2(3x) - sin^3(3x) + 2cos(x)cos(3x)sin(3x) = sin(7x)
Now, we can work on solving this equation. However, it is important to note that this equation may not have a simple algebraic solution. It might require numerical methods or approximations.
If you are looking to find the values of x that satisfy this equation, you can use numerical methods. One approach is to plot both sides of the equation and find the points where they intersect. Alternatively, you can use software or calculators that have built-in equation solvers.
It's also worth mentioning that there can be multiple solutions to this equation, so it's important to consider the domain you are interested in.
Keep in mind that manual methods might not be practical for this particular equation. Using numerical methods or appropriate software will likely yield better results.