# Sinusoidal Graph question

• Sep 27th 2012, 08:43 PM
Higg
Sinusoidal Graph question
Hey everyone, I'm at a question that I'm a little stumped on the method of solving, and there is no solution guide in the textbook :(.

4. In a New Zealand city the time of sunrise can be found using the trigonometric function

T= 4.5sin(pi(d-4)/180)+4.9

where T is the number of hours past midnight d and is the number of days into the year. Find the time to the nearest minute when the
Sun will rise on the 7th day of the year. How many days in the year does the sunrise after 7 am?

The bolded part of the question is the part I am not sure how to solve.

Any help? :D
• Sep 27th 2012, 08:55 PM
MaxJasper
Re: Sinusoidal Graph question
Days of the year when sunrise is after 7 a.m. are:

{32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156}

which are 125 days.

http://mathhelpforum.com/attachment....1&d=1348808188

This can't be NZ(Shake)
• Sep 27th 2012, 09:11 PM
MarkFL
Re: Sinusoidal Graph question
You want to solve:

$4.5\sin\left(\frac{\pi(d-4)}{180} \right)+4.9>7$

$4.5\sin\left(\frac{\pi(d-4)}{180} \right)>2.1$

$\sin\left(\frac{\pi(d-4)}{180} \right)>\frac{7}{15}$

So we want:

$\sin^{-1}\left(\frac{7}{15} \right)<\frac{\pi(d-4)}{180}<\pi-\sin^{-1}\left(\frac{7}{15} \right)$

$\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)

$\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)+4

You should find, using your calculator that (approximately):

$31.8

Since d is an integer, we may write:

$32\le d\le156$

Hence, the number of days N is:

$N=(156-32)+1=125$