Question

5sin(x/2)=cos(x/2)cos(x/4)

Answer 1

Для этой задачи можно начертить только график

Answer 2

To solve the trigonometric equation 5sin(x/2) = cos(x/2)cos(x/4), we will use trigonometric identities to simplify the equation and find the possible solutions.

Step 1: Use double-angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1

Step 2: Rewrite the equation using the double-angle identity for cos(x/2): 5sin(x/2) = 2cos^2(x/2) - 1

Step 3: Rewrite sin(x/2) in terms of cos(x/2) using the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1 sin(θ) = √(1 - cos^2(θ))

Since we have sin(x/2) and cos(x/2), we can rewrite sin(x/2) as follows: sin(x/2) = √(1 - cos^2(x/2))

Step 4: Replace sin(x/2) in the equation with the new expression: 5√(1 - cos^2(x/2)) = 2cos^2(x/2) - 1

Step 5: Square both sides of the equation to eliminate the square root: (5√(1 - cos^2(x/2)))^2 = (2cos^2(x/2) - 1)^2

Step 6: Simplify and solve for cos(x/2): 25(1 - cos^2(x/2)) = 4cos^4(x/2) - 4cos^2(x/2) + 1

Step 7: Rearrange the equation and collect terms: 4cos^4(x/2) - 4cos^2(x/2) + 1 - 25 + 25cos^2(x/2) = 0

Step 8: Combine like terms: 4cos^4(x/2) + 21cos^2(x/2) - 24 = 0

Now, we have a quadratic equation in terms of cos^2(x/2). Let's make a substitution to simplify it further:

Let u = cos^2(x/2)

The equation becomes: 4u^2 + 21u - 24 = 0

Step 9: Factor the quadratic equation: (4u - 3)(u + 8) = 0

Now, set each factor equal to zero and solve for u:

1. 4u - 3 = 0 4u = 3 u = 3/4

2. u + 8 = 0 u = -8 (We ignore this solution since cos^2(x/2) cannot be negative.)

Step 10: Substitute back for u to find cos^2(x/2): cos^2(x/2) = 3/4

Step 11: Find cos(x/2) by taking the square root of both sides: cos(x/2) = ±√(3/4) = ±√3/2

Now, we have two possible values for cos(x/2): cos(x/2) = √3/2 and cos(x/2) = -√3/2.

Step 12: Find the possible solutions for x/2:

1. If cos(x/2) = √3/2: x/2 = π/6 + 2πn (where n is an integer) x = π/3 + 4πn

2. If cos(x/2) = -√3/2: x/2 = 5π/6 + 2πn (where n is an integer) x = 5π/3 + 4πn

So, the solutions for the given equation are: x = π/3 + 4πn x = 5π/3 + 4πn

where n is an integer.