Question

Sin(2x)*sin(6x)=cos(x)*cos(3x) I need some help with it)))))

Answer 1

$sin2x\cdot sin6x=cosx\cdot cos3x\\\\\frac{1}{2}\cdot (cos4x-cos8x)=\frac{1}{2}\cdot (cos4x+cos2x)\\\\cos2x+cos8x=0\\\\2\, cos5x\cdot cos3x=0\\\\a)\; \; cos5x=0\; ,\; \; 5x=\frac{\pi }{2}+\pi n,\; \; \underline {x=\frac{\pi }{10}+\frac{\pi n}{5}\; ,\; n\in Z}\\\\b)\; \; cos3x=0\; ,\; \; 3x=\frac{\pi }{2}+\pi k\; ,\; \; \underline {x=\frac{\pi}{6}+\frac{\pi k}{3}\; ,\; k\in Z}$

Answer 2

Sure, I'd be happy to help you with this trigonometric equation!

Let's start by using some trigonometric identities to simplify the equation. We'll use the following identities:

1. Product-to-Sum Identity: sin(a) * sin(b) = (1/2) * [cos(a - b) - cos(a + b)]
2. Double-Angle Identity: sin(2x) = 2 * sin(x) * cos(x)
3. Triple-Angle Identity: cos(3x) = 4 * cos^3(x) - 3 * cos(x)

Now, let's proceed with the solution:

Given equation: sin(2x) * sin(6x) = cos(x) * cos(3x)

Using the double-angle identity for sin(2x): 2 * sin(x) * cos(x) * sin(6x) = cos(x) * cos(3x)

Divide both sides by cos(x) (assuming cos(x) ≠ 0): 2 * sin(x) * sin(6x) = cos(3x)

Using the product-to-sum identity for sin(a) * sin(b): (1/2) * [cos(x - 6x) - cos(x + 6x)] = cos(3x)

Simplify the angles inside the cosines: (1/2) * [cos(-5x) - cos(7x)] = cos(3x)

Using the identity: cos(-θ) = cos(θ) (1/2) * [cos(5x) - cos(7x)] = cos(3x)

Now, let's focus on simplifying the right-hand side (cos(3x)) using the triple-angle identity: cos(3x) = 4 * cos^3(x) - 3 * cos(x)

Our equation now becomes: (1/2) * [cos(5x) - cos(7x)] = 4 * cos^3(x) - 3 * cos(x)

To solve this equation, we'll move all the terms to one side: (1/2) * [cos(5x) - cos(7x)] - 4 * cos^3(x) + 3 * cos(x) = 0

Now, we have an equation in terms of cos(x) and its multiples. We can proceed to solve it using numerical methods or graphical methods.

If you need an exact solution, it's unlikely that it can be found using elementary algebraic methods, as it involves trigonometric functions with different angles. However, if you need an approximate numerical solution, you can use numerical techniques like numerical root-finding methods (e.g., Newton-Raphson method) or graphing the functions to find the points where they intersect.

If you have specific values for x or any other constraints, please provide them, and I can help you further with the calculations.