2sin5xcos4x-sinx=sin9x
Can u do this without triple angle formula and how
Yes, it is possible to solve this equation without using the triple angle formula. Here's how you can approach it:
1. Start by simplifying the equation to eliminate any trigonometric terms with higher angles. In this case, we can rewrite the equation as:
2sin(5x)cos(4x) - sin(9x) = 0
2. Next, try to factor out a common term that will allow you to simplify the equation further. In this case, notice that sin(5x) is a common factor in the first two terms:
sin(5x)(2cos(4x) - 1) - sin(9x) = 0
3. Now, we have two trigonometric terms: sin(5x) and sin(9x). To simplify further, we can use the sum-to-product formulas, which state that sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2). Applying this formula, we can rewrite the equation as:
sin(5x)(2cos(4x) - 1) - 2sin((5x+9x)/2)cos((5x-9x)/2) = 0
Simplifying further:
sin(5x)(2cos(4x) - 1) - 2sin(7x)cos(-2x) = 0
sin(5x)(2cos(4x) - 1) + 2sin(7x)cos(2x) = 0
4. Now, we can factor out a common term of sin(5x) to simplify the equation even more:
sin(5x)(2cos(4x) - 1 + 2cos(2x)) = 0
5. At this point, we have a product of two factors that equals zero. According to the Zero Product Property, if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for x:
a) sin(5x) = 0
To find the solutions to this equation, we can set 5x equal to the values for which sin(θ) = 0:
5x = 0, π, 2π, ...
This gives us the solutions:
x = 0, π/5, 2π/5, ...
b) 2cos(4x) - 1 + 2cos(2x) = 0
To solve this equation, we can simplify it further:
2cos(4x) + 2cos(2x) = 1
cos(4x) + cos(2x) = 1/2
Using the double angle formula, we can rewrite cos(2x) as 2cos^2(x) - 1:
cos(4x) + 2cos^2(x) - 1 = 1/2
Rearranging the terms:
2cos^2(x) + cos(4x) = 3/2
Now, we need to graph the function y = 2cos^2(x) + cos(4x) - 3/2 and find the x-values where the graph intersects y = 0.
Note: I'm sorry, but as a text-based AI, I can't graph functions for you. You can use online graphing tools or computer software to plot the graph and determine the solutions for cos(4x) + 2cos^2(x) = 3/2.
Once you find the x-values where cos(4x) + 2cos^2(x) = 3/2, you can solve for x using algebraic methods.
These are the steps you can follow to solve the given equation without using the triple angle formula.