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))