To determine which equation also models the temperature [tex]\( t \)[/tex] as a function of months [tex]\( m \)[/tex], let's start with the given function:
[tex]\[ t = 35 \cos \left(\frac{\pi}{6}(m+3)\right) + 55 \][/tex]
We need to find an equivalent equation using the sine function. Let's rewrite the cosine function in terms of the sine function using trigonometric identities. We can use the identity:
[tex]\[ \cos(x) = \sin\left( \frac{\pi}{2} - x \right) \][/tex]
First, let's set [tex]\( x = \frac{\pi}{6}(m + 3) \)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{6}(m + 3)\right) \][/tex]
Using the identity, we substitute [tex]\( x \)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{6}(m + 3)\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{6}(m+3)\right) \][/tex]
Now, simplify the argument of the sine function:
[tex]\[ \frac{\pi}{2} - \frac{\pi}{6}(m + 3) = \frac{\pi}{2} - \frac{\pi}{6}m - \frac{\pi}{6} \cdot 3 \][/tex]
[tex]\[ = \frac{\pi}{2} - \frac{\pi}{6}m - \frac{\pi}{2} \][/tex]
[tex]\[ = -\frac{\pi}{6}m \][/tex]
Thus:
[tex]\[ \cos\left(\frac{\pi}{6}(m + 3)\right) = \sin\left(-\frac{\pi}{6}m\right) \][/tex]
Next, we use the odd property of the sine function, which states:
[tex]\[ \sin(-x) = -\sin(x) \][/tex]
So:
[tex]\[ \sin\left(-\frac{\pi}{6}m\right) = -\sin\left(\frac{\pi}{6}m\right) \][/tex]
Substituting this back into the original equation:
[tex]\[ t = 35 \cos\left(\frac{\pi}{6}(m + 3)\right) + 55 \][/tex]
[tex]\[ = 35 \left[\sin\left(-\frac{\pi}{6}m\right)\right] + 55 \][/tex]
[tex]\[ = 35 \left[-\sin\left(\frac{\pi}{6}m\right)\right] + 55 \][/tex]
[tex]\[ = -35 \sin\left(\frac{\pi}{6}m\right) + 55 \][/tex]
Thus, the equivalent equation that also models the average daily temperature [tex]\( t \)[/tex] as a function of months [tex]\( m \)[/tex] is:
[tex]\[ t = -35 \sin\left(\frac{\pi}{6}m\right) + 55 \][/tex]
So, the correct answer is:
[tex]\[ t = -35 \sin\left(\frac{\pi}{6}m\right) + 55 \][/tex]
