Tuesday, September 9, 2014

Trigonometric Formulas Geometrically

Trigonometry comes from the Greek trigōnon ("triangle") and metron ("measurement").
Geometry comes from the Greek geo ("earth") and metron ("measurement"). 

The Greek mathematicians Euclid and Archimedes in the 3rd century BC were the first to prove trigonometric formulas geometrically. You can find the values of sine, cosine and tangent functions, by using a unit circle with a radius = 1. This article will explain how to develop Hip -Valley Rafter Roof Ratios using  Trigonometric Formulas geometrically drawn using Tetrahedrons or Trirectangular Tetrahedrons for the Trigonometric Ratios we use in roof framing. The tetrahedron unfolded is the same as the roof framing kernel. The trirectangular tetrahedron unfolded is a tetrahedron slice from a compound joint. 

The "Unit Circle" is a circle with a radius of 1.



For a triangle with an angle θ, the trigonometric ratios functions are calculated this way:
Sine Function:
sin(θ) = Opposite / Hypotenuse
Cosine Function:
cos(θ) = Adjacent / Hypotenuse
Tangent Function:
tan(θ) = Opposite / Adjacent
For a triangle with an angle θ, the trigonometric ratios functions are calculated this way for roof framing:
Sine Function:
sin(θ) = Rafter Rise / Rafter Length
Cosine Function:
cos(θ) = Rafter Run / Rafter Length
Tangent Function:
tan(θ) = Rafter Rise / Rafter Run


Because the radius is 1, you can directly measure sine, cosine and tangent.



Because the radius is 1, you can also directly measure cosecant, secant and cotangent.
Because the radius is 1, you can also directly measure the Trigonometric Ratios of an polygon. An Octagon in this example.

Basic Roof Framing Tetrahedron unfolded.
D Angle = Plan Angle
A Angle = Roof Slope Angle
C Angle = Hip Rafter Slope Angle
E Angle = Jack Rafter Side Cut Angle
B Angle = Hip Rafter Backing Angle


In this example the triangle in blue has the following properties:
Roof Slope Angle = ß  = 33.69007°
Plan Angle = α = 45.00°
Radius = 1

The altitude of the triangle can be found by multiplying the sine of the triangle by the cosine of the triangle. 
Altitude of Triangle =  sin ß × cos ß
sin ß = 0.5547
cos ß  = 0.8321
Altitude of Triangle =  sin ß × cos ß = 0.5547 × 0.8321 = 0.4615





In this example the triangle in blue has the following properties:
Hip Rafter Slope Angle = ß  = 25.23940°
Plan Angle = α = 45.00°
Radius = 1
 Again, the altitude of the triangle can be found by multiplying the sine of the triangle by the cosine of the triangle. 
Altitude of Triangle =  sin ß × cos ß
sin ß =0.4264
cos ß  =  0.9045 
Altitude of Triangle =  sin ß × cos ß = 0.4264 × 0.9045 = 0.3857




If were studying tetrahedrons for roof framing trigonometric ratios we could also use following trigonometric ratios to find the base length of the plan angle triangle or the height of the plan angle triangle.
cos α × cos ß =  base length of the plan angle triangle
cos α × cos ß = 0.6396
0.7071 × 0.9045 = 0.6396

sin α × cos ß = height of the plan angle triangle
sin α × cos ß = 0.6396
0.7071 × 0.9045 =0.6396


Since the Altitude of Triangle =  sin ß × cos ß
we can use the following Trigonometric Ratios to write an formula for the hip rafter backing angle when the plan angle is equal to 45.00°.

Hip Rafter Backing Angle = 
arctan((sin ß × cos ß) ÷ cos ß) =
arctan((sin ß × cos ß) ÷ cos ß) =
arctan(sin ß)  =

arctan(sin 25.23940°) = 23.09347°
arctan(sin(Hip Rafter Slope Angle)) = Hip Rafter Backing Angle



When the plan angle is not equal to 45.00° you need to use the tan of the plan angle in the trigonometric formula for the hip rafter backing angle.

Hip Slope Angle = ß  =  27.50055°
Plan Angle = α = 51.34019°
Radius = 1

Hip Rafter Backing Angle = 
arctan((sin ß × cos ß) ÷ (cos ß × tan α)) =
arctan((sin ß × cos ß) ÷ (cos ß × tan α)) =
arctan(sin ß ÷ tan α) =
arctan(sin 27.50055° ÷ tan 51.34019) = 20.27452°
arctan(sin Hip Rafter Slope Angle ÷ tan Plan Angle)= Hip Rafter Backing Angle



In this example you can find the trigonometric ratios to write a formula for the Hip Rafter Slope Angle, angle C in the unfolded tetrahedron.  You have to extend the base of triangle A, Rafter Slope Angle Triangle, so the Hypotenuse (Rafter Length) is equal to 1. So, you can dimension the correct length of the cosine of the Rafter Slope Angle.

Angle C = arccos( cos( A ) ÷ cos ( B ))
or
Hip Rafter Slope Angle = arccos( cos( Roof Slope Angle ) ÷ cos (Hip Rafter Backing Angle ))
cos A = 0.83205
cos B = 0.93804
cos C = ( 0.83205 ÷ 0.93804)
cos C = ( 0.88701)
Angle C = arccos( 0.83205 ÷ 0.93804) =27.50055
Angle C = arccos(0.88701) =27.50055



It's useful to print out the Unit Circle Trigonometric Functions-Ratios for the triangles you're studying, when you're trying to write roof framing formulas.  This way you can try different combinations of the trigonometric ratios using multiplication or division. 
Example using   multiplication of the cosine of D and the values of angle A:
0.62470 × 0.83205 = 0.51978 ...no match
0.62470 ×  0.55470 = 0.34652 ...matches the sin of the angle B
0.62470 ×  0.66667 = 0.41647 ...no match

From these matches we can write
Angle B = arcsin(cos(D) × sin(A))
Hip Rafter Backing Angle = arcsin(cos(Plan Angle) × sin(Roof Slope Angle)) 


D Angle = 51.34019 = Plan Angle
cos = 0.62470
sin = 0.78087
tan = 1.24999

A Angle = 33.69007 = Roof Slope Angle
cos = 0.83205
sin = 0.55470
tan = 0.66667

C Angle = 27.50055 = Hip Rafter Slope Angle
cos = 0.88701
sin = 0.46176
tan = 0.52058

E Angle = 33.64933 = Jack Rafter Side Cut Angle
cos = 0.83244
sin = 0.55411
tan = 0.66564

B Angle = 20.27452 = Hip Rafter Backing Angle
cos = 0.93804
sin = 0.34652
tan = 0.36941


Here's an example for writing an trigonometric formula for the tetrahedron angle E, jack rafter side cut angle.
Angle E = arctan(cos(A) ÷ tan(D))
Angle E = arctan(0.83205 ÷ 1.24999)
Angle E = arctan(0.66564) = 33.64933 
Jack Rafter Side Cut Angle = arctan(cos(Roof Slope Angle) ÷ tan( Plan Angle))



Here's an example of using the values of tan, cosine and sine of a tetrahedron to write a trigonometric for the Angle D in the tetrahedron. (D = Plan Angle)
Angle D = arccos(sin(E) ÷ cos(C))
Angle D = arccos(0.55411 ÷ 0.88701)
Angle D = arccos(0.62470) = 51.34019
Plan Angle = arccos(sin(Jack Rafter Side Cut Angle) ÷ cos( Hip Rafter Slope Angle))


3 comments:

  1. Sim,
    This is a great article! I've been trying to prove to myself the deceivingly simple formula for the bevel angle of crown molding that originally appeared in a Fine Homebuilding article (FHB #68, pp. 79-81). The formula is essentially the same as arcsine(cos(Plan Angle) × sin(Roof Slope Angle)) for the hip backing angle, except that the actual values of the crown drop/the crown width are substituted for the sine of the roof slope angle.
    Your proved that the product of those two values is the hip backing angle by trial and error. I’m still having problems developing the formula geometrically and/or algebraically. Can you help? Thanks!

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    Replies
    1. Chris,
      For the crown molding bevel angle

      arcsine(cos(Plan Angle) × sin(90-Roof Slope Angle)) = bevel angle
      or
      arcsine(cos(Plan Angle) × cos(Roof Slope Angle)) = bevel angle

      I'm not familiar with this.
      crown drop/the crown width are substituted for the sine of the roof slope angle

      developing the formula geometrically
      Yeah, I had a PDF file that shows how to draw out the crown bevels using a framing square, but I can't find it right now.
      Sim

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    2. Ok. Let me take a step back to the unit circle you presented above. How is it that Angle C = arccos( cos( A ) ÷ cos ( B ))? My gut feeling is that somehow tan (D) cancels out, but I'm not sure of that. Is this right?

      Delete