Question

Cos п(2x-1)/3=1/2 в ответ запишите наибольший отрицательный корень

Answer 1

$Cos\frac{\pi(2x-1) }{3} =\frac{1}{2}\\\\\frac{\pi(2x-1) }{3}=\pm arcCos\frac{1}{2}+2\pi n,n\in Z\\\\\frac{\pi(2x-1)}{3}=\pm\frac{\pi }{3}+2\pi n,n\in Z\\\\2x-1=\pm1+6n,n\in Z\\\\\left \{ {{2x=2+6n} \atop {2x=6n,n\in Z}} \right.\\\\\left \{ {{x=1+3n} \atop {x=3n,n\in Z}} \right.$

При n ≥ 0 будем получать только положительные корни, значит :

1) n = - 1    ⇒    x = 1 + 3 * (- 1) = - 2

2) n = - 1   ⇒    x = 3 * (- 1) = - 3

Ответ : - 2

Answer 2

To find the largest negative root for the equation:

Cos[(2x-1)/3] = 1/2

Let's solve the equation step by step:

Step 1: Take the inverse cosine (arccos) of both sides to remove the cosine function:

arccos(Cos[(2x-1)/3]) = arccos(1/2)

Step 2: Since the range of arccos is [0, π], we need to consider the positive and negative values for 1/2:

2x - 1 = ±π/3

Step 3: Solve for x in each case:

Case 1: 2x - 1 = π/3 2x = π/3 + 1 x = (π/3 + 1)/2

Case 2: 2x - 1 = -π/3 2x = -π/3 + 1 x = (-π/3 + 1)/2

Now, let's check which value corresponds to the largest negative root.

Since π/3 is approximately 1.0472 and 1 is a positive number, the value (π/3 + 1)/2 is positive.

For -π/3, which is approximately -1.0472, (π/3 - 1)/2 is negative. So, the largest negative root is:

x ≈ (-π/3 + 1)/2 ≈ (-(-1.0472) + 1)/2 ≈ 0.5236

The largest negative root of the equation is approximately 0.5236.