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------------------------------------------------ If [tex]\tan(t) = 57[/tex], what is [tex]\tan(t-\pi)[/tex]?

A. -57
B. 57
C. -\tan(t)
D. \tan(t)

Given that [tex]tan(t)=57[/tex] the value of [tex]tan(t-π)[/tex] is [tex]-57[/tex], considering the properties of the tangent function's periodicity and its odd nature.

Explanation:

To determine the value of tan(t-π) given that [tex]tan(t)=57[/tex], we can use the periodic properties of the tangent function. The tangent function has a period of π which means that [tex]tan(θ) = tan(θ ± nπ)[/tex] for any integer n. Therefore [tex]tan(t-π) = tan(t)[/tex] because subtracting [tex]π[/tex] from t is equivalent to moving one full period along the tangent function.

However it is also important to recognize that the tangent function, while periodic, is odd. This fact means that [tex]tan(-θ) = -tan(θ)[/tex]. When subtracting [tex]π[/tex] from t in [tex]tan(t-π)[/tex], we are essentially finding the tangent of the angle that is π radians less than t which is equivalent to finding the tangent of [tex]-t[/tex]. Therefore,[tex]tan(t-π) = -tan(t) = -57[/tex].