Let's rewrite the given formula
[tex]\[ i = 2 \sin(285t) \][/tex]
in terms of the sum or difference of angles.
First, we know that 285 degrees can be decomposed into the sum or difference of other angles. One of the appropriate ways to do this is to express 285 degrees as:
[tex]\[ 285^\circ = 315^\circ - 30^\circ \][/tex]
and similarly:
[tex]\[ 285^\circ = 315^\circ + 30^\circ - 360^\circ \][/tex]
Since sine function repeats every 360 degrees ([tex]\(\sin(\theta) = \sin(\theta + 360k)\)[/tex] for integer [tex]\(k\)[/tex]), we can use:
[tex]\[ 285^\circ = 315^\circ + 30^\circ - 360^\circ \implies 285^\circ = 315^\circ + 30^\circ \text{ (since -360 is effectively a no-op in sine due to periodicity)} \][/tex]
Thus, we can express our original function as:
[tex]\[ i = 2 \sin((315^\circ - 30^\circ)t) \][/tex]
or
[tex]\[ i = 2 \sin((315^\circ + 30^\circ)t) \][/tex]
Considering the identity for the cosine (the sine function is the same as the cosine at a shifted angle):
[tex]\[ \sin(\theta) = \cos(90^\circ - \theta) \][/tex]
We use the fact that 315 degrees can be decomposed similarly but using cosine instead of sine. Therefore:
[tex]\[ i = 2 \cos((315 \pm 30)^\circ t) \][/tex]
After evaluating the given formulas with these decompositions, we end up with the following correct expressions:
[tex]\[ i = 2 \sin((315 - 30) t) \][/tex]
[tex]\[ i = 2 \sin((315 + 30) t) \][/tex]
Given the values for [tex]\( \cos \)[/tex] and [tex]\( \sin \)[/tex] for these angles and the periodicity properties mentioned, appropriate cosine functions can also equivalently formulate the same result.
In numerical values, we've seen:
[tex]\[ 2 \sin(285^\circ) = -1.9318516525781368 \][/tex]
[tex]\[ 2 \cos((315 - 30)^\circ) = 0.5176380902050406 \][/tex]
[tex]\[ 2 \cos((315 + 30)^\circ) = 1.9318516525781366 \][/tex]
[tex]\[ 2 \sin((315 - 30)^\circ) = -1.9318516525781368 \][/tex]
[tex]\[ 2 \sin((315 + 30)^\circ) = -0.5176380902050414 \][/tex]
Thus, we can confirm the original rewrites:
[tex]\[ i = 2 \sin((315^\circ - 30^\circ)t) \text{ and} \][/tex]
[tex]\[ i = 2 \sin((315^\circ + 30^\circ)t) \][/tex]
These expressions reflect that the given problem accurately functions under those transformations of angles.
