To determine which equation also models the average daily temperature, we start by analyzing the given equation for temperature [tex]\( t \)[/tex]:
[tex]\[ t = 35 \cos \left( \frac{\pi}{6}(m+3) \right) + 55 \][/tex]
We will examine each of the given options to see if any of them are equivalent to the above equation.
Option 1: [tex]\( t = -35 \sin \left( \frac{\pi}{6} m \right) + 55 \)[/tex]
Rewrite the given equation in a different form using trigonometric identities. We need to express it in terms of sine, knowing that cosine can be converted to sine using the identities:
[tex]\[ \cos x = \sin \left( \frac{\pi}{2} - x \right) \][/tex]
So,
[tex]\[ t = 35 \cos \left( \frac{\pi}{6}(m+3) \right) + 55 \][/tex]
Using the trigonometric identity:
[tex]\[ \cos \left( \frac{\pi}{6}(m+3) \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}(m+3) \right) \][/tex]
Simplify the argument:
[tex]\[ \cos \left( \frac{\pi}{6}(m+3) \right) = \sin \left( \frac{\pi}{2} - \left( \frac{\pi}{6}(m+3) \right) \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}m - \frac{3\pi}{6} \right) = \sin \left( \frac{\pi}{2} - \frac{\pi}{6}m - \frac{\pi}{2} \right) = \sin \left( -\frac{\pi}{6}m \right) \][/tex]
We know:
[tex]\[ \sin(-x) = -\sin(x) \][/tex]
Thus:
[tex]\[ \sin \left( -\frac{\pi}{6}m \right) = -\sin \left( \frac{\pi}{6}m \right) \][/tex]
We then rewrite the equation as:
[tex]\[ t = 35 \cdot (-\sin \left( \frac{\pi}{6}m \right)) + 55 = -35 \sin \left( \frac{\pi}{6}m \right) + 55 \][/tex]
This matches Option 1 perfectly.
Hence, the equation that also models the average daily temperature 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]
