There are 3 basic trig functions; sin(), cos() and tan() (see Trigonometric Functions http://en.wikipedia.org/wiki/Cosine). These functions describe the ratios of the sides of a right triangle. Here's a right triangle with an angle of 30° (this would be the side view of a cone with angle of repose of 30°).

            t .|
         o .   |
      p .      | opposite (height)
   y .         | 
h .  30)       | 
 --------------
   adjacent (radius)

The definition of these 3 trig functions is based on a right triangle, and is

sin() = opposite/hypoteneuse
cos() = adjacent/hypoteneuse
tan() = opposite/adjacent

The tan() is commonly known as the slope (in this case of the hypoteneuse). In a right triangle, once you've chosen two of the sides, the other side is determined (by Pythagorus), so once you know the value for any of these functions for any angle, the value for the other functions is determined.

Probably the best known trig function is the sin() (see the red and green graphs in the section Unit circle definitions http://en.wikipedia.org/wiki/Cosine#Unit-circle_definitions, labelled "sin and cos in the Cartesian plane"), also see Sine wave http://en.wikipedia.org/wiki/Sine_wave). The sine wave describes oscillatory phenomena and the projection of an object moving in a circle, e.g. the position of a pendulum, the pressure in a sound wave, the electric field in an electromagnetic wave (e.g. radio, light), the position of the tip of a propellor.

What is sin() for the triangle below with an angle of 30°, which will have the dimensions as shown 90° ^[376] ?

            t .|
length=2 o .   |
      p .      | opposite, length=1
   y .         | 
h .  30)       | 
 --------------
   adjacent

What is the length of the adjacent side ^[377] ?

Now that you know the lengths of all of the sides of this triangle, what is cos(30), tan(30) ^[378] ?

By drawing the appropriate triangle, find tan(45) ^[379] ?

What is sin() for the triangle below as the angle approaches 0°, 90° ^[380] ?

            t .|
         o .   |
      p .      | opposite (height)
   y .         | 
h .  angle)    | 
 --------------
   adjacent (radius)

While sin(),cos() vary continuously, tan() reaches ∞ at 90°.

Any math calculator will have these functions. The argument (i.e. the angle) that the functions take, can be in degrees (the everyday unit) or in radians (the mathematician's unit). You can flip from one measure to the other: a circle has 360° or 2π radians.

Radian measure is convenient for many functions involving trig. The "2π" come about because a circle has a circumference C=2πr. A semi-circle has a circumference of πr. This allows us to define an angle as the ratio of the arc of the circumference divided by the radius. With a circle having 2π radians, then a quadrant of a circle (90°) subtends an angle at the centre of π/2 (pronounced "π on 2").

     .
    |   .
    |     .
    |      .
    |pi/2   .
    --------

A mathematician cutting up a pie for desert, will ask the other mathematicians what they want. They'll reply "I'd like π on 2" or if they're not particularly hungry "I'd like π on 3 please". Any non-mathematician at the dinner, not understanding the difference between pie and π might be alarmed at hearing several requests for "π on 2", thinking there will be no pie left.

Python's default unit for angles is radians

igloo:# python
Python 2.4.4 (#3, Mar 30 2007, 19:33:13) 
[GCC 3.4.6] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from math import *
>>> radians(45)
0.78539816339744828
>>> tan(radians(45))
0.99999999999999989

Using python, find sin(30), cos(60), sin(0), cos(0) ^[381] .

The discovery, through calculus, of quickly converging series to calculate sin(),cos() and tan(), were a great aid to surveyors, who changed the social structure of Europe by dividing it into owned plots of land. The US president George Washington was a surveyor by trade.

The distance from the earth to the sun is 146Mm. The diameter of the sun is 1.38Mm. What is the apparent diameter of the sun in degrees? Assume the size of the sun is the length of the opposite side, while the distance to the sun is the adjacent side. First find the tan() of the apparent diameter of the sun. Here's what the calculation looks like in python

dennis:/src/da/python_class# python
Python 2.4.3 (#1, Apr 22 2006, 01:50:16) 
[GCC 2.95.3 20010315 (release)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from math import *
>>> 1.38/146
0.009452054794520548 	#tan() of the angular diameter of the sun
>>> atan(1.38/146)
0.0094517733231954185	#the angle (in radians) of the diameter of the sun
>>> degrees(atan(1.38/146))
0.54154672033343809 	#the diameter of the sun in degrees
>>> 60*degrees(atan(1.38/146))
32.492803220006287 	#the diameter of the sun in minutes

diameter of sun=32.49min

Doing it this way, the triangle isn't quite a right triangle (the angle is 89.75°, which is close enough for what we're doing). If the angle wasn't so small we'd do it this way.

                  .|
              .    | d
          .        | i 
      .            | a
oo)----------------  m 
      .dist to sun |
          .        | s 
               .   | u
                  .| n

The diagram shows a viewer looking at the sun, with the line to the center of the sun and the diameter making a right triangle. You first calculate the radius of the sun (half the diameter) and calculate the angle subtended by half the sun. You double this to get angle subtended by the sun.

>>> 1.38/(146*2)
0.004726027397260274	#tan of half the diameter of the sun
>>> atan(1.38/(146*2))
0.0047259922119301497	#angular size of half of the sun, radians
>>> degrees(atan(1.38/(146*2)))
0.27077940775529408	#angular size of half of the sun, degrees
>>> 2*degrees(atan(1.38/(146*2)))
0.54155881551058815	#angular size of the whole sun, degrees
>>> 60*2*degrees(atan(1.38/(146*2)))
32.493528930635293	#angular size of the sun, minutes

diameter of sun=32.49min

For small angles, the sloppy and the exact result are the same.

A handy reference for locating stars, planets and constellations in the sky, or the distance between mountains or ships on the horizon, is the distance across the knuckles of your hand when your arm is extended. This angular distance doesn't change much as the body grows and changes size. Measure the distance from your shoulder to your knuckles, then the distance across your knuckles. Find the (approximate) angle between your knuckles as viewed by your eye ^[382] .

I have relatively small hands. The usual figure quoted is 8°. The distance between pairs of knuckles is 2°. How many sun widths can you fit between a pair of knuckles ^[383] ? You can fit 12-16 sun diameters across your outstretched knuckles.

Example: you're standing on with a clear view of the horizon and see the sun in the west about two hand (i.e. across the knuckles) widths above the horizon. Actually you need to measure the distance along the line of the ecliptic, which being inclined to the vertical, will be a bit longer than the vertical, but lets ignore this detail for this exercise. How long is it till sunset ^[384] ? It's mid september. What time is it ^[385] ?

Example: The diameter of the moon is 3474km and its distance from the earth varies from 363,104km (apogee) to 405,696km (perigee) (see Moon http://en.wikipedia.org/wiki/Moon). What is the range of the angular size of the moon (in minutes) ^[386] ?

The moon varies about 10% in diameter during its orbit about the earth (you won't notice this variation by casual observation). The moon rotates at constant angular velocity about its own axis, but because of the elliptical orbit, the speed at which the moon moves through the sky varies through the month. and the moon appears to oscillate from side to side about its axis, called libration, presenting more than half the moon's surface to viewers on the earth (we can see about 4/7 of the surface of the moon over a month). For an animation of libration see Libration (http://en.wikipedia.org/wiki/Libration) (note the prominent crater Tycho, http://en.wikipedia.org/wiki/Tycho_(crater) in the southern highlands). For view of the moon's differing size in the same orientation, see shots of the moon taken over a period of 2 years Libration: 2 years in 2 seconds (http://pixheaven.net/voir_us.php?taille=grand&mon=0505-0704).

By a fantastic coincidence, the moon is 400 times closer than the sun and is 400 times smaller than the sun. This means that the moon can exactly eclipse the sun. Because of the ellipticity of the moon's orbit, a viewer on the earth's surface can see a total lunar eclipse or an annular lunar eclipse. A total lunar eclipse occurs when the moon covers the face of the sun. An annular lunar eclipse occurs when the moon doesn't quite cover the face of the sun and a ring of the sun is seen around the moon. Because of the angular size of the moon and sun are so close, the sun's annulus only shows between the mountains of the moon, leading to a series of bright points around the moon, known as Baily's Beads (http://en.wikipedia.org/wiki/Baily's_beads). Here's the angular size of the sun and the moon at apogee/perigee

diameter of sun                    32.49min
diameter of moon at apogee         29.44min
diameter of moon at perigee        32.99min

Is it possible to have a total lunar eclipse, an annual lunar eclipse at apogee, perigee ^[387] ? Note how at perigee the moon is only just big enough to fully cover the sun's face. Most total eclipses are annular.

You'll be lucky to see a total solar eclipse once in your life. The path is only a couple of 100m wide, so you have to be exactly at the right spot to see Bailey's Beads (offer your assistance to a bunch of astronomers - they'll know where to go). The eclipse I saw was on 30 May 1984. We followed the weather forecast from a wet and cloudy Maryland, phoning a friend at NOAA in Colorado every couple of hours for updates - as to how far south we'd have to go (this is before the internet and before weather.com). We had to go about 2 states further south than we'd expected, to set up our telescopes in a large empty parking lot outside a church under a brilliant clear sky, in Georgia (we'd phoned ahead for permission). We set up a line of about 8 telescopes equipped with TV cameras, at about 10m intervals (to make sure at least one of them saw Bailey's Beads), transverse to the path and let the cameras roll. To synchronise the time (again before the internet), we used radios tuned to the time signal at WWV, and let the TV camera microphone pick up the time signals. The leader of the party set up in the middle of the line, to get the best view of Baily's beads, but his calculations were off by about 10m and the next telescope in the line got the best view.

The next total solar eclipse in NA Solar Eclipse 21 Aug 2017 (http://eclipse.gsfc.nasa.gov/SEanimate/SEanimate2001/SE2017Aug21T.GIF). passes right over our heads here in NC. It will be a long eclipse (by eclipse standards) - 2min:43sec (the peak for this eclipse is 2:45 just a bit west of us). All your friends will be coming to see it. People you'd forgot were your friends will be arriving to see you. Don't miss it! (Let's hope it's a clear day.)

Let's find a few angles in the Pyramid of Khufu. Here's an ascii art diagram of the pyramid from above (the plan view http://en.wikipedia.org/wiki/Plan_view) and from the side (the elevation view http://en.wikipedia.org/wiki/Elevation_(view)) looking up one of the edges of the pyramid.

b         a
-----------                 x 
| .      .|                /|\     ^
|   .  .  |               / | \    |
|    x    |l=h*pi/2      /  |  \   h
|  .   .  |             /   |   \  |
|.       .|            / A) |    \ v
-----------            -----------
a         b            a    b    a

"A" is the angle between an edge and the base. We know that the (total length of the sides/height)=2π. This gives the length of each side. What is angle that each edge makes with the base (labelled "A" in the right hand diagram)? Hint - first determine tan(A). More hints ^[388] Here's the answer ^[389] .

The limestone casing stones were moved to the face of the pyramid as rectangular blocks, and the cut and polished in place. The builders used a jig to determine the angle of the face. Here's how it would have been done The Tura stone quarries (http://www.cheops-pyramide.ch/khufu-pyramid/stone-quarries.html#tura)

Now let's calculate the angle between the faces and the base. Once the angle between the edge and the base is determined, is angle between the base and the face fixed? (Can you make different pyramids all with the same angle between the edge and the base ^[390] ? Can you prove this ^[391] ?)

Here's the diagram of the pyramid again, this time with the elevation view rotated by 45°. "A" is the angle between a face and the base.

-----------                 x
| .      .|                / \     ^
|   .  .  |               /   \    |
|    x    |l=h*pi/2      /     \   h
|  .   .  |             /       \  |
|.       .|            / A)      \ v
-----------            -----------
a         b            a         b

As well as the length of the side and the height of the pyramid, we now also know the length of the diagonal across the base. How do we determine tan(A) (Hint: ^[392] )? Here's the diagram with the hint added ^[393] . Here's a much too complicated way of finding the length of the adjacent side (this was my first method, only read this if you have lots of time) ^[394] , but here's a much simpler way of doing it ^[395] . Here's the answer ^[396] .

The Egyptians didn't measure the slope (the tan() of the angle that the surface makes), but the inverse of the slope (which in English is sometimes called the inclination, a term that has many meanings, one of which is the angle of departure from the vertical). Instead of measuring the rise in height with a fixed distance (the slope), Egyptians measured the distance required to raise the height by a fixed amount. This is the same as the slope of a rod which is normal to the face. This the Egyptians called the seked (pronounced seqt; see Ancient Egytpian units of measurement http://en.wikipedia.org/wiki/Ancient_Egyptian_weights_and_measures).

	Note
	Also called the "seqt". See Thales (http://en.wikipedia.org/wiki/Thales) where Thales measures the height of the pyramids by measuring the length of the shadow at a time when his shadow was the same as his height. In this URL see a worked problem on the slope of the sides of a pyramid.

In current terminology the seked is cot()=1/tan()=tan(co-angle=90°-angle). A seked of 5 (a horizontal distance of 5 palms, per cubit of rise) is 54.46°, 5&1/4 seked (5 palms and 1 digit) is 53.13°, and 5&1/2 seked (5 palms and 2 digits) is 51.84°. It would seem that the face of the Great Pyramid has a seked of 5&1/2.

Ralph Greenberg explains in Pi and the Great Pyramid (http://www.math.washington.edu/~greenber/PiPyr.html) that just by good luck π≅22/7 and that a choice of inclination of 5&1/2 seked gives a slope of 28/22. Greenberg says that the origin of dividing cubits into 7 intervals is lost in the past, at least back to 150yrs earlier than the Great Pyramid to the era of Imhotep (http://en.wikipedia.org/wiki/Imhotep) who is considered to be the first engineer known by name.

Here we see that the ratio of 22/7 to π is out by 0.04%, the same error as the ratio (distance around the base)/height.

dennis:# echo "(22/7)/(4*a(1))" | bc -l	#(22/7)/pi
1.00040249943477068197

It's a little disappointing to find that with the precision of the other measurements of the Great Pyramid, that the Egyptians were using 22/7 for their calculations and not π.

What is the relationship between the angle base-to-edge and the angle base-to-face? Assume that the length of one side of the base is l rather than h*π/2. Find the formulae for the two angles. Here's my answer ^[397] .

If tan(edge-to-base)=x, then tan(face-to-base)=x*sqrt(2) (the sqrt(2) is the ratio of the length of the hypoteneuse to the sides in a 45° right triangle). Since the two angles are different, for what angle(s) are they most different? With a flat pyramid, both angles will be nearly the same. With a very high pyramid, both angles will also be nearly the same. We can guess that the two angles will have the greatest difference somewhere in between (say with the angles about 45°). I decided to find this out for fun; It's not designed to illuminate any computing or trigonometric principle. Here's a graph with edge-to-base angle on the x-axis and face-to-base on the y-axis (code is here ^[398] ).

Figure 12. graph of the angles in a pyramid: red line x-axis: angle edge to base. y-axis: angle face-to-base. grey line - reference line where both angles are equal.

The maximum difference can be found in a couple of equivalent ways

The ratio of the tan() should be sqrt(2).

pip:~# python
Python 2.4.4 (#2, Mar 30 2007, 16:26:42) 
[GCC 3.4.6] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> sqrt(2)
1.4142135623730951	#what we want
>>> tan(radians(40))
0.83909963117727993
>>> tan(radians(50))
1.19175359259421
>>> tan(radians(50))/tan(radians(40))
1.4202766254612063	#not quite correct
.			#several trials
.
>>> tan(radians(49.94))/tan(radians(40.06))
1.4142495433444027	#close enough

It would seem that the maximum difference occurs somewhere close to (x,y)=(40.06°,49.94°)

Its dimensions of the Transamerica Pyramid (from http://ecow.engr.wisc.edu/cgi-bin/get/cee/340/bank/11studentpro/transamericabuilding-2.doc) are 174'*174'*853'. The bottom 5 stories have vertical walls; presumably the pyramid starts at the 6th floor. Let's assume it's a pyramid. What is the angle edge-to-base, and face-to-base ^[399] ?