The average temperature for the city of Madison, in degrees Fahrenheit, is described by the function [math]t(m) = 25 \sin [\frac{\pi}{6}(m - 3)]+50[/math], where [math]m[/math] is the number of months that have elapsed since January 1. In what month will the average temperature reach 60°F? What day, approximately? Assume the year is not a leap year.
[list=1] [*]Set up the equation to be solved. [*]Isolate the trigonometric function. [*]Use the appropriate inverse function to cancel out the trigonometric function. [*]Evaluate the term containing the inverse trigonometric function, making sure your answer is within the range of the function. [*]Expand the answer using your knowledge of reference angles and periodicity to find all possible solutions. [*]Solve the results for [math]m[/math]. [*]Test that the results satisfy the original equation. [*]Use the results to determine when the average temperature will be 60°F. [/list]