We know that the cosine function has a maximum value of 1 and a minimum value of -1. Since both equations are equal to -1, we can infer that the arguments of the cosine function are such that they correspond to the maximum value of the function.
For cos(π√(x+1)) = -1, the argument π√(x+1) must correspond to the value of π, because cos(π) = -1.
Hence, we can write:
π√(x+1) = π
Solving for x: √(x+1) = 1 x + 1 = 1 x = 0
So, the solution to the first equation is x = 0.
Now, for cos(π√(x-4)) = -1, the argument π√(x-4) must again correspond to π, because cos(π) = -1.
Hence, we can write:
π√(x-4) = π
Solving for x: √(x-4) = 1 x - 4 = 1 x = 5
Therefore, the solution to the second equation is x = 5.
In conclusion, the solutions to the given equations are x = 0 and x = 5.
To solve the given equations:
cos(π√(x+1)) = -1cos(π√(x-4)) = -1We know that the cosine function has a maximum value of 1 and a minimum value of -1. Since both equations are equal to -1, we can infer that the arguments of the cosine function are such that they correspond to the maximum value of the function.
For cos(π√(x+1)) = -1, the argument π√(x+1) must correspond to the value of π, because cos(π) = -1.
Hence, we can write:
π√(x+1) = π
Solving for x:
√(x+1) = 1
x + 1 = 1
x = 0
So, the solution to the first equation is x = 0.
Now, for cos(π√(x-4)) = -1, the argument π√(x-4) must again correspond to π, because cos(π) = -1.
Hence, we can write:
π√(x-4) = π
Solving for x:
√(x-4) = 1
x - 4 = 1
x = 5
Therefore, the solution to the second equation is x = 5.
In conclusion, the solutions to the given equations are x = 0 and x = 5.