Question

Cos(2x+п/4)+cosx/2=0

Answer 1

Вот ответ. Надеюсь, поможет :D

Answer 2

To solve the trigonometric equation cos(2x + π/4) + cos(x/2) = 0, we'll need to find the values of x that satisfy this equation. Let's break it down step by step:

Step 1: Combine the two cosine terms: cos(2x + π/4) + cos(x/2) = 0

Step 2: Use the trigonometric identity for the sum of cosines: cos(A) + cos(B) = 2*cos((A + B)/2)*cos((A - B)/2)

Applying this identity to our equation: 2*cos((2x + π/4 + x/2)/2) * cos((2x + π/4 - x/2)/2) = 0

Step 3: Simplify the arguments inside the cosine functions: (2x + π/4 + x/2)/2 = (5x + π)/4 (2x + π/4 - x/2)/2 = (3x + π)/4

So, the equation becomes: 2*cos((5x + π)/4) * cos((3x + π)/4) = 0

Step 4: Set each cosine factor to zero and solve for x:

1. cos((5x + π)/4) = 0
2. cos((3x + π)/4) = 0

For (1), let's solve for x: (5x + π)/4 = π/2 + kπ where k is an integer. 5x + π = 2π + 4kπ 5x = π + 4kπ x = π/5 + 4kπ/5

For (2), let's solve for x: (3x + π)/4 = π/2 + kπ where k is an integer. 3x + π = 2π + 4kπ 3x = π + 4kπ x = π/3 + 4kπ/3

So the general solutions for x are:

1. x = π/5 + 4k*π/5
2. x = π/3 + 4k*π/3

where k is any integer (positive, negative, or zero).

These are the values of x that satisfy the given trigonometric equation. Keep in mind that there are infinitely many solutions, as there are infinitely many integers for k.