Question

Sin6x*cosx+cos6x*sinx=1/2 Помогите.

Answer 1

Решение...................

Answer 2

Solving the Trigonometric Equation

To solve the trigonometric equation sin(6x)cos(x) + cos(6x)sin(x) = 1/2, we can use trigonometric identities and properties to simplify the equation and find the solution.

Trigonometric Identities and Properties

We can start by using the trigonometric identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B). This identity allows us to rewrite the left-hand side of the equation in a more simplified form.

Applying Trigonometric Identity

Using the identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B), we can rewrite the left-hand side of the equation as: sin(6x + x) = sin(7x).

Simplified Equation

Therefore, the equation becomes: sin(7x) = 1/2.

Finding the Solution

To find the solution for sin(7x) = 1/2, we can use the inverse sine function or arcsin. The solution for sin(7x) = 1/2 is: 7x = π/6 + 2πn or 7x = 5π/6 + 2πn, where n is an integer.

Final Solution

Dividing both sides by 7, we get: x = π/42 + (2π/7)n or x = 5π/42 + (2π/7)n, where n is an integer.

This provides the general solution for the trigonometric equation sin(6x)cos(x) + cos(6x)sin(x) = 1/2.