Question

Sin4x+корень3cos4x=0

Answer 1

Решение на фотографии

Answer 2

To solve the trigonometric equation Sin(4x) + √3cos(4x) = 0, we can use trigonometric identities and algebraic manipulation. First, let's rewrite √3cos(4x) as (√3/2) * 2cos(4x). This will help us combine the terms effectively.

The equation becomes:

Sin(4x) + (√3/2) * 2cos(4x) = 0

Now, we can use the identity sin(θ) + cos(θ) = √2 * sin(θ + π/4) to simplify further:

√2 * sin(4x + π/4) = 0

Now, we have:

sin(4x + π/4) = 0

To solve this equation, set the argument of sin to zero:

4x + π/4 = k * π

where k is an integer. Now, isolate x:

4x = k * π - π/4

x = (k * π - π/4)/4

This gives us the general solution for x in terms of k. We can plug in different integer values for k to find multiple solutions. For example:

1. When k = 1: x = (1 * π - π/4)/4 = (3π - π)/4 = (2π)/4 = π/2

2. When k = 2: x = (2 * π - π/4)/4 = (7π - π)/4 = (6π)/4 = 3π/2

So, the solutions for the equation Sin(4x) + √3cos(4x) = 0 are x = π/2 + 2πn and x = 3π/2 + 2πn, where n is an integer (n = 0, ±1, ±2, ...). These solutions represent the values of x that satisfy the original trigonometric equation.