# Help with general trig equation

• Mar 28th 2010, 11:48 AM
baconfoot
Help with general trig equation
I don't know what this problem would be classified as, but i need help with it!

The formula F(x)=75+15sin[2π/365(x-101)] represents the fahrenheit temperature in Gainesville as a function of the day of the year. Let x represent the day of the year (in a non leap year), so x=1 represents Jan. 1 (etc.) Use this formula to determine the day(s) of the year (give the dates) when the average daily temperature will be closest to 85ºF

I graphed it in a graphing calculator and determined that the days were day 143 and day 240 (April 23rd and July 28) and plugged them into the equation and they equaled about 85, but i do not know how to work the problem out using trig functions.

Any help would be appreciated. Thank you!
• Mar 28th 2010, 11:53 AM
e^(i*pi)
Quote:

Originally Posted by baconfoot
I don't know what this problem would be classified as, but i need help with it!

The formula F(x)=75+15sin[2π/365(x-101)] represents the fahrenheit temperature in Gainesville as a function of the day of the year. Let x represent the day of the year (in a non leap year), so x=1 represents Jan. 1 (etc.) Use this formula to determine the day(s) of the year (give the dates) when the average daily temperature will be closest to 85ºF

I graphed it in a graphing calculator and determined that the days were day 143 and day 240 (April 23rd and July 28) and plugged them into the equation and they equaled about 85, but i do not know how to work the problem out using trig functions.

Any help would be appreciated. Thank you!

Work backwards and use arcsin (inverse sine)

$\displaystyle \frac{F(x) - 75}{15} = \sin \left(\frac{2\pi}{365}(x-101)\right)$

$\displaystyle \arcsin \left(\frac{F(x) - 75}{15}\right) = \left(\frac{2\pi}{365}(x-101)\right)$

$\displaystyle x-101 = \frac{365}{2\pi} \times \arcsin \left(\frac{F(x) - 75}{15}\right)$

$\displaystyle x = 101 + \frac{365}{2\pi} \times \arcsin \left(\frac{F(x) - 75}{15}\right)$
• Mar 28th 2010, 12:44 PM
baconfoot
hey thanks so much!
but how do I get to 240? I tried to use the reference angle, but apparently thats not right.