Solve for x (exact solutions):
sin x - sin 3x + sin 5x = 0
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Thankyou. I tried find the solutions on my graphics calculator but however, the question is to find the EXACT solution(s) for the equation so I guess it will be in a fractional form or a square root form. Any help is appreciated of course.
Use these identities and see what you get
sin 3x = 3 sinx - 4 sin^3x
sin 5x = 5 sinx- 20sin^3x + 16sin^5x
This means that
3 sinx + 16 sin^3x + 16 sin^5x = 0
or
sin x*(16 sin^4x + 16 sin^2x +3 = 0)
One solution is sinx = 0; x = 0.
The other solution can be obtained by letting sin^2x = u and solving the quadratic equation
16 u^2 + 16 + 3 = 0
This can be factored into
(4u + 3)(4u + 1) = 0
To solve this equation, we can use the properties of trigonometric functions and identities. Let's start by simplifying the equation using the sum-to-product trigonometric identity.
The sum-to-product trigonometric identity states that sin(A) + sin(B) = 2*sin((A+B)/2)*cos((A-B)/2). Therefore, we can rewrite the equation as:
sin(x) + sin(5x) = 2*sin((x+5x)/2)*cos((x-5x)/2)
sin(x) + sin(5x) = 2*sin(3x)*cos(-2x)
Since cos(-2x) = cos(2x), we can simplify the equation further:
sin(x) + sin(5x) = 2*sin(3x)*cos(2x)
Now, let's use the product-to-sum trigonometric identity. The product-to-sum identity states that sin(A)*cos(B) = (1/2)*(sin(A+B) + sin(A-B)). Applying this identity to our equation, we get:
sin(x) + sin(5x) = sin(3x)*cos(2x) + sin(3x)*cos(-2x)
sin(x) + sin(5x) = sin(3x)*(cos(2x) + cos(-2x))
Again, since cos(-2x) = cos(2x), we can simplify further:
sin(x) + sin(5x) = 2*sin(3x)*cos(2x)
Now, the equation can be rewritten as:
sin(x) + sin(5x) - 2*sin(3x)*cos(2x) = 0
This is now a single trigonometric equation. To solve it, we need to find the values of x that satisfy this equation.
Unfortunately, there is no general algebraic method to find the exact solutions to this trigonometric equation. Instead, we can use numerical methods or a calculator to approximate the solutions.