# Thread: Daylight problem using derivatives of sine/cosine

1. ## Daylight problem using derivatives of sine/cosine

The length of daylight is approx a sinusoidal function of the day of the yr. The longest day is june 21, the 172 day of the year. The longest day has 14 hr 3 min of daylight. The shortest is 10 hr 15 min. Write equation to express the length of daylight in min/day as a function of day of the year. How much daylight is there on august 7?

2. ## Re: Daylight problem using derivatives of sine/cosine

Is this the equation? y=2916+228pi/86(d-172)

3. ## Re: Daylight problem using derivatives of sine/cosine

This thread solves a similar problem.

4. ## Re: Daylight problem using derivatives of sine/cosine

Originally Posted by mockingjay95
Is this the equation? y=2916+228pi/86(d-172)
This expression is lacking the sine.

5. ## Re: Daylight problem using derivatives of sine/cosine

Sorry, my bad, is it y=2916+228cospi/86(d-172)?

6. ## Re: Daylight problem using derivatives of sine/cosine

Originally Posted by mockingjay95
is it y=2916+228cospi/86(d-172)?
First, please you parentheses. It is not clear whether cospi/86(d-172) means cos(pi)/86(d-172), cos(pi/86)(d-172) or cos(pi/86(d-172)). Second, the function cos(pi/86(d-172)) has a period of 2 * 86 = 172 days instead of 365 days. Indeed, cos(pi/86((d + 172) - 172)) = cos(pi/86(d - 172) + pi/86 * 172) = cos(pi/86(d - 172) + 2pi) = cos(pi/86(d - 172)). Finally, y(172) = 2916 + 228 = 3144 whereas 14h 3m is 843 m.

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### trig daylight problem

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