Question

Sin2x+sin8x-sin3x-sin7x=0 :) ​

Answer 1

Ответ:

Скрин

Объяснение:

Answer 2

To solve the trigonometric equation sin(2x) + sin(8x) - sin(3x) - sin(7x) = 0, we'll use some trigonometric identities to simplify the equation and then solve for x.

Step 1: Use the angle sum and difference identities to rewrite the equation. The angle sum and difference identities for sine are as follows:

• sin(A + B) = sin(A) * cos(B) + cos(A) * sin(B)
• sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)

Step 2: Apply the identities to our equation. sin(2x) + sin(8x) - sin(3x) - sin(7x) = 0 Using the identities, we can rewrite the equation as follows: (sin(2x) - sin(7x)) + (sin(8x) - sin(3x)) = 0

Step 3: Use another trigonometric identity. Another identity that will be helpful is the sum-to-product identity:

• sin(A) - sin(B) = 2 * cos((A + B)/2) * sin((A - B)/2)

Step 4: Apply the sum-to-product identity to the equation. (sin(2x) - sin(7x)) + (sin(8x) - sin(3x)) = 0 Using the sum-to-product identity, we can rewrite it as: 2 * cos((2x + 7x)/2) * sin((2x - 7x)/2) + 2 * cos((8x + 3x)/2) * sin((8x - 3x)/2) = 0

Simplify the equation further: 2 * cos(9x/2) * sin(-5x/2) + 2 * cos(11x/2) * sin(5x/2) = 0

Step 5: More simplification using a trigonometric identity. Another identity that can help is the double angle identity for sine:

• sin(2A) = 2 * sin(A) * cos(A)

Step 6: Apply the double angle identity to the equation. 2 * cos(9x/2) * sin(-5x/2) + 2 * cos(11x/2) * sin(5x/2) = 0 Using the double angle identity for sine, we can rewrite it as: 2 * cos(9x/2) * [2 * sin(-5x/4) * cos(-5x/4)] + 2 * cos(11x/2) * [2 * sin(5x/4) * cos(5x/4)] = 0

Step 7: Simplify further. 2 * cos(9x/2) * [2 * (-sin(5x/2))] + 2 * cos(11x/2) * [2 * sin(5x/2)] = 0

Now, the equation is much simpler: -4 * cos(9x/2) * sin(5x/2) + 4 * cos(11x/2) * sin(5x/2) = 0

Step 8: Factor out the common term (sin(5x/2)). 4 * sin(5x/2) * [cos(11x/2) - cos(9x/2)] = 0

Step 9: Set each factor to zero and solve for x.

1. sin(5x/2) = 0 To solve sin(5x/2) = 0, we get: 5x/2 = nπ, where n is an integer. x = (2nπ)/5

2. cos(11x/2) - cos(9x/2) = 0 To solve cos(11x/2) - cos(9x/2) = 0, we get: cos(11x/2) = cos(9x/2)

Using the identity: cos(A) = cos(B) => A = ±B + 2πn, where n is an integer. We have two cases: a. 11x/2 = 9x/2 + 2πn x = 2π*n (One set of solutions)

b. 11x/2 = -9x/2 + 2πn 20x/2 = 2πn x = π*n (Another set of solutions)

So the complete solution set for the given trigonometric equation is: x = (2nπ)/5 or x = (nπ) for any integer n.