1. ## 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?

2. ## 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.

This can't be NZ

3. ## Re: Sinusoidal Graph question

You want to solve:

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

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

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

So we want:

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

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

$\displaystyle \frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)+4<d<184-\frac{180}{\pi}\sin^{-1}\left(\frac{7}{15} \right)$

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

$\displaystyle 31.8<d<156.2$

Since d is an integer, we may write:

$\displaystyle 32\le d\le156$

Hence, the number of days N is:

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