To determine which of the given options are equivalent to the function [tex]\( y = 3 \cos x + 2 \)[/tex], let’s analyze each option step-by-step.
### Option A: [tex]\( y = 3 \sin \left(x - \frac{\pi}{2}\right) + 2 \)[/tex]
We know the trigonometric identity:
[tex]\[ \sin \left( x - \frac{\pi}{2} \right) = -\cos(x) \][/tex]
So, substituting into the equation:
[tex]\[ y = 3 \sin \left(x - \frac{\pi}{2}\right) + 2 = 3(-\cos(x)) + 2 = -3 \cos(x) + 2 \][/tex]
This is clearly not equivalent to [tex]\( y = 3 \cos x + 2 \)[/tex].
Therefore, Option A is not equivalent.
### Option B: [tex]\( y = 3 \sin \left(x + \frac{\pi}{2}\right) + 2 \)[/tex]
We know the trigonometric identity:
[tex]\[ \sin \left( x + \frac{\pi}{2} \right) = \cos(x) \][/tex]
So, substituting into the equation:
[tex]\[ y = 3 \sin \left(x + \frac{\pi}{2}\right) + 2 = 3 \cos(x) + 2 \][/tex]
This is exactly the given function [tex]\( y = 3 \cos x + 2 \)[/tex].
Therefore, Option B is equivalent.
### Option C: [tex]\( y = 3 \cos (-x) + 2 \)[/tex]
We know that the cosine function is an even function, which means:
[tex]\[ \cos(-x) = \cos(x) \][/tex]
So, substituting into the equation:
[tex]\[ y = 3 \cos(-x) + 2 = 3 \cos(x) + 2 \][/tex]
This is exactly the given function [tex]\( y = 3 \cos x + 2 \)[/tex].
Therefore, Option C is equivalent.
### Option D: [tex]\( y = -3 \cos x - 2 \)[/tex]
Clearly, this equation has the opposite signs for the coefficients of both the [tex]\(\cos(x)\)[/tex] term and the constant term. It does not match the given function [tex]\( y = 3 \cos x + 2 \)[/tex].
Therefore, Option D is not equivalent.
### Conclusion:
After analyzing all the options, we find that:
- Options B and C are equivalent to [tex]\( y = 3 \cos x + 2 \)[/tex].
- Options A and D are not equivalent to [tex]\( y = 3 \cos x + 2 \)[/tex].
Thus, the correct options are B and C.
