Calculating Daylight Hours Using Trigonometry

  • Thread starter TonyC
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In summary, the number of hours of daylight in a town on the west coast of North America can be modeled by the formula h=3.75sin[(2pi/365)(d-79)] + 12, where h is the number of hours of daylight in a day and d is the day of the year. Using this formula, the total accumulated number of hours of daylight between March 29 and June 29 is 1355.6 hours. However, this formula may need to be adjusted to ensure all days have a real number of hours. Another formula using a sum of sines can be used to calculate the total hours of sunlight from day n to day N.
  • #1
TonyC
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The number of hours of daylight in a town on the west coast of North America can be modeled by:

h=3.75sin[2pi/365(d-79)] + 12

Where h is the number of hours of daylight in a day and d is the day of the year, with d=1 representing January 1 (assume 28 days in Feb). What is the total accumlated number of hours of daylight by a town between Mar 29 and June 29.

I worked the problem and came up with 1355.6 hours.

Am I good to go?
 
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  • #2
You proberlly ment the formula more like this: h=3.75sin[(2pi/365)(d-79)] + 12. In the previous form it would be undefined for the 79th day and as far as I know all days have a real number of hours. As for your resoult I can't even imagine a way to get more than 15.75 hours. You probelly just made a mistake when tiping it in your calculator. I would sugest taht you do it in steps and not all at once :wink: .
 
  • #3
I don't know how you got your result but I wrote a program to work it out!

Taking March 29 as day 88 and June 29 as day 180, I got the following,

day 88 to day 180 inclusive: total hrs = 1369.77
day 89 to day 180 inclusive: total hrs = 1357.18
 
  • #4
You can write a general formula by summing the sequence of sines. Use Euler's formula for the trig functions and sum it as geometric series.
 
  • #5
The hours of sunlight from day n to day N inclusive is:

[tex]12(N - n + 1) + 3.75\sum _{d = n} ^{d = N}\sin \left (\frac{2\pi (d - 79)}{365}\right ) = 12(N - n + 1) + 3.75\sum _{d = n - 79} ^{N - 79}\sin \left (\frac{2\pi d}{365}\right )[/tex]
 
  • #6
This may help:

[tex]\sum_{n = a}^{b} \sin nx = \frac { \cos x(a-1/2) - \cos x(b+1/2)} {2 \sin x/2}[/tex]

(Edited: YIKES! Sorry, akg! I went dyslexic when I typed it and interchanged the x with the a and b!)
 
Last edited:
  • #7
Wait a minute, what if a = b = 1?

sin(x) = [cos(x - 1/2) - cos(x + 1/2)]/2sin(x/2)
= [cos(x)cos(1/2) + sin(x)sin(1/2) - cos(x)cos(1/2) + sin(x)sin(1/2)]/2sin(x/2)

= sin(x)sin(1/2)/sin(x/2)
sin(x/2) = sin(1/2)

That doesn't seem right. Where did you get that formula from?
 
  • #8
It seems right now.

I used tide's sum of sines formula in akg's expression for hours of sunlight and got the same results as my computer.
 

What is Trig and what does it stand for?

Trig refers to trigonometry, which is a branch of mathematics that deals with the relationship between the sides and angles of triangles. It comes from the Greek words "trigonon" meaning triangle and "metron" meaning measure.

Why is Trig important?

Trig is important because it has a wide range of applications in fields such as engineering, physics, astronomy, and navigation. It helps us understand and solve problems involving angles and triangles, which are fundamental concepts in many areas of science and technology.

What are the basic trigonometric functions?

The basic trigonometric functions are sine, cosine, and tangent. These functions relate the ratios of the sides of a right triangle to its angles.

How do you use Trig in real life?

Trigonometry is used in various real-life situations, such as calculating the height of a building, determining the distance between two points, and designing bridges and other structures. It is also used in navigation, astronomy, and in the study of sound and light waves.

What are some common formulas used in Trig?

Some common formulas used in Trig include the Pythagorean theorem, which relates the lengths of the sides of a right triangle, and the law of sines and law of cosines, which are used to solve triangles with known angles and sides. Other important formulas include the double and half angle formulas, which are used to simplify trigonometric expressions.

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