Question

Найдите корень уравнения cos пи(х-1)/3=1/2. В ответ запишите напишите наибольший отрицательный корень.

Answer 1

Cos π(x-1)/3 = 1/2
π(x-1)/3=+-π/3+2πk
x-1=3(+-π/3)+6πk
x-1=+-π+6πk
x=1+-π+6πk
k=0  x=1-π≈1-3.14=-2.14
k=1  6π-π+1 >0
k= -1  -6π+π+1< -2.14
наибольший отрицательный корень 1-π

Answer 2

Solving the Equation

To find the roots of the equation cos(pi(x-1)/3) = 1/2, we need to solve for x. The equation can be rewritten as:

cos(pi(x-1)/3) - 1/2 = 0

To solve this equation, we can use the trigonometric identity cos(2θ) = 2cos^2(θ) - 1. Applying this identity, we have:

2cos^2(pi(x-1)/6) - 1 - 1/2 = 0

Simplifying further:

2cos^2(pi(x-1)/6) - 3/2 = 0

Now, let's solve for cos^2(pi(x-1)/6):

cos^2(pi(x-1)/6) = 3/4

Taking the square root of both sides:

cos(pi(x-1)/6) = ±sqrt(3)/2

Now, we can solve for pi(x-1)/6:

pi(x-1)/6 = arccos(±sqrt(3)/2)

To find the values of x, we need to consider the range of the inverse cosine function. The range of arccosine is [0, pi]. Therefore, we have two cases to consider:

1. pi(x-1)/6 = arccos(sqrt(3)/2) 2. pi(x-1)/6 = arccos(-sqrt(3)/2)

Case 1: pi(x-1)/6 = arccos(sqrt(3)/2)

Solving for x in this case:

pi(x-1)/6 = arccos(sqrt(3)/2)

Multiplying both sides by 6/pi:

x-1 = (6/pi) * arccos(sqrt(3)/2)

Adding 1 to both sides:

x = 1 + (6/pi) * arccos(sqrt(3)/2)

Case 2: pi(x-1)/6 = arccos(-sqrt(3)/2)

Solving for x in this case:

pi(x-1)/6 = arccos(-sqrt(3)/2)

Multiplying both sides by 6/pi:

x-1 = (6/pi) * arccos(-sqrt(3)/2)

Adding 1 to both sides:

x = 1 + (6/pi) * arccos(-sqrt(3)/2)

Finding the Largest Negative Root

To find the largest negative root, we need to evaluate the expressions for x in both cases and select the largest negative value.

Evaluating the expression for x in Case 1:

x = 1 + (6/pi) * arccos(sqrt(3)/2)

Evaluating the expression for x in Case 2:

x = 1 + (6/pi) * arccos(-sqrt(3)/2)

To determine which value is the largest negative root, we need to evaluate the expressions numerically. However, without the exact value of pi, we cannot provide the specific numerical solution.

Please note that the exact value of pi is an irrational number and cannot be expressed as a finite decimal. Therefore, we cannot provide the exact numerical value of the largest negative root without knowing the value of pi.

If you have the exact value of pi, you can substitute it into the expressions for x in both cases and compare the results to find the largest negative root.