Question

Cos(x/2)=0,5 розвяжить бистро

Answer 1

$\cos( \frac{x}{2} ) = 0.5 \\ \cos( \frac{x}{2} ) = \frac{1}{2} \\ \frac{x}{2} = ± \arccos( \frac{1}{2} ) \\ \frac{x}{2} = ± \frac{\pi}{3} + 4\pi n \\ x = ± \frac{2\pi}{3} + 4\pi n, \: n \in Z$

Answer 2

To solve the equation cos(x/2) = 0.5, we can follow these steps:

Step 1: Get the half-angle identity for cosine: cos(x/2) = ±√((1 + cos(x))/2)

Step 2: Set cos(x/2) = 0.5: 0.5 = ±√((1 + cos(x))/2)

Step 3: Square both sides to remove the square root: 0.5^2 = ((1 + cos(x))/2)

Step 4: Simplify: 0.25 = (1 + cos(x))/2

Step 5: Multiply both sides by 2 to get rid of the fraction: 2 * 0.25 = 1 + cos(x)

Step 6: Subtract 1 from both sides: 0.5 = cos(x)

Now, to find the solutions for x, we need to find the angles where cos(x) equals 0.5. The cosine function has a period of 2π (or 360 degrees), and it repeats its values every 2π. The cosine function takes on the value of 0.5 at two specific angles: 60 degrees (π/3 radians) and 300 degrees (5π/3 radians). Also, it takes the value of 0.5 at their coterminal angles: 420 degrees (7π/3 radians) and -60 degrees (-π/3 radians).

Therefore, the solutions for x are: x = π/3 + 2πn (where n is an integer) or x = 5π/3 + 2πn (where n is an integer)

where n represents any integer, and it allows us to find all possible solutions in the domain of real numbers.