Question

Cos (п/2-x/2) -3 cos (п-x/2)=0 Решите пожалуйста.

Answer 1

Sin(x/2) + 3cos(x/2) =0
sin(x/2)   +   3cos(x/2) =    0         
cos(x/2)        cos(x/2)     cos(x/2)
tg(x/2) +3 =0
tg(x/2) = -3
x/2 = -arctg3 +πk, k∈Z
x= -2arctg3 + 2πk, k∈Z

Answer 2

$cos( \frac{ \pi }{2} - \frac{x}{2} )-3cos( \pi - \frac{x}{2} )=0$

$sin \frac{x}{2} +3cos \frac{x}{2} =0$

$\frac{sin \frac{x}{2} }{cos \frac{x}{2} } +\frac{3cos \frac{x}{2} }{cos \frac{x}{2} } =0$

$\frac{sin \frac{x}{2} }{cos \frac{x}{2} } +3=0$

$tg \frac{x}{2} =-3$

$\frac{x}{2} =-arctg3+ \pi n x=-2arctg3+2 \pi n$

Answer 3

Problem Statement

We are given the equation cos(π/2 - x/2) - 3cos(π - x/2) = 0 and we need to solve for x.

Solution

To solve the equation, let's simplify it step by step.

Step 1: Simplify the equation using trigonometric identities. Using the identity cos(π - θ) = -cos(θ), we can rewrite the equation as: cos(π/2 - x/2) + 3cos(x/2) = 0.

Step 2: Apply the double-angle formula for cosine. The double-angle formula for cosine is cos(2θ) = 2cos^2(θ) - 1. Using this formula, we can rewrite the equation as: 2cos^2(x/2) - 1 + 3cos(x/2) = 0.

Step 3: Simplify the equation further. Let's substitute y = cos(x/2). Then the equation becomes: 2y^2 - 1 + 3y = 0.

Step 4: Solve the quadratic equation. To solve the quadratic equation, we can factor it or use the quadratic formula. Factoring the equation, we get: (2y + 1)(y - 1) = 0.

Setting each factor equal to zero, we have two possible solutions: 1. 2y + 1 = 0: Solving for y, we get y = -1/2.

2. y - 1 = 0: Solving for y, we get y = 1.

Step 5: Solve for x. Now that we have the values of y, we can substitute them back into the equation y = cos(x/2) to solve for x.

For y = -1/2: cos(x/2) = -1/2.

For y = 1: cos(x/2) = 1.

To find the values of x, we need to consider the range of the cosine function. The cosine function has a range of [-1, 1].

For cos(x/2) = -1/2: The solutions for x/2 are 7π/6 + 2πn and 11π/6 + 2πn, where n is an integer.

For cos(x/2) = 1: The solutions for x/2 are 0 + 2πn and 2π + 2πn, where n is an integer.

Therefore, the solutions for x are: - x = 7π/3 + 4πn, - x = 11π/3 + 4πn, - x = 2πn, - x = 4π + 2πn, where n is an integer.

Please note that these solutions are in radians. If you need the solutions in degrees, you can convert them using the conversion factor: 180°/π.

I hope this helps! Let me know if you have any further questions.