Math Notes #1

Roof Eave Angle = 90.00000
SS = Main Rafter Slope Angle = 36.86990
S = Adjacent Rafter Slope Angle = 45.00000
DD = Main Plan Angle = 53.13010
D = Adjacent Plan Angle = 36.86990
R1 = Hip Rafter Slope Angle = 30.96376
C5m Main Hip Rafter Backing Angle = 21.10020
C5a Adjacent Hip Rafter Backing Angle = 34.44990
Valley Sleeper Saw Blade Bevel Angle = 90° - (C5m + C5a) = 34.44990
P2m = Main Jack Rafter Side Cut Angle = 30.96376
90° - P2m = Main Roof Sheathing Angle = 59.03624
P2a = Adjacent Jack Rafter Side Cut Angle = 43.31386
90° - P2a = Adjacent Roof Sheathing Angle = 46.68614

R8-DP = Tilted Hip Rafter Slope Angle on DP Side = arctan( tan ( Hip Rafter Backing Angle) ÷ tan (Jack Rafter Side Cut Angle ))

C8 = (C5m + C5a) = 55.55010
R8-DP = arctan( tan ( C5m )  ÷  tan ( P2m ))
R8-DP = arctan( tan ( 21.10020 )  ÷  tan ( 30.96376 )) = 32.74587

R9-DP = Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Footprint to Hip Rafter Run Line
R9-DP = arctan( sin ( R1 ) × tan ( C5m ))
R9-DP = arctan( sin ( 30.96376 ) × tan ( 21.10020 )) = 11.22889

D-DP = Horizontal Plane Rotation Angle for Tilted Hip Rafter on DP Line from Eave Line to Hip Rafter Footprint
D-DP = D - R9
D-DP = 36.86990 - 11.22889 = 25.64101

D-DP = arctan((cos SS ÷ tan DD) ÷ (1 ÷ cos SS))
D-DP = arctan((cos 36.86990 ÷ tan 53.13010) ÷ (1 ÷ cos 36.86990)) = 25.64101

R10-DP = Vertical Plane Rotation Angle from Plumb for Tilted Hip Rafter on DP Line
R10-DP = arctan((cos(R1) × tan(R9)) / sin(R1));
R10-DP = arctan((cos(30.96376) × tan(11.22889)) / sin(30.96376)) = 18.30847

Rotation Angle = 70°
S = Adjacent Rafter Slope Angle = 45.00000
Rotated Rafter Slope Angle = arctan(tan(Rafter Slope Angle) × sin(Rotation angle))
Rotated Rafter Slope Angle = arctan(tan(45.00°) × sin(70°)) = 43.219179°

sin 18.30847° = b = 0.314132

90° - R9-DP = C

90° - 11.22889° = C = 78.77708°

A = D -( 90° - Rotation Angle)

A = 36.86990° -( 90° - 70)= 16.86990°

B = 180° - A - C = 84.35302°

Law of Sines c = ( b × sin( C ) ) ÷ sin( B)

c = ( 0.314132 × sin( 78.77708° ) ) ÷ sin( 84.35302°) = 0.30962837

Rafter Vertical Plane Rotation = arctan(c ÷ cos R10-DP)

Rafter Vertical Plane Rotation = arctan(0.30962837 ÷ cos 18.30832° ) = 18.06258°

Rafter Upper Claw Angle = (90° - Rafter Slope Angle) - Rafter Vertical Plane Rotation

Rafter Upper Claw Angle = (90° - 43.219179°) - 18.06258° = 28.71824°

Long version of trigonometry for the upper claw angle

R10-DP = arctan((cos(30.96376) × tan(11.22889)) / sin(30.96376)) = 18.30847

90° - R10-DP = Vertical Rotate Angle

90° - R10-DP = 71.69168°

(Plan Angle + Purlin Rotation Angle) - 90° = Horizontal Rotate Angle

(36.86990° + 70°) - 90° = 16.86990°

90° - 18.30832 = 71.69168°

cos 71.69168° = 0.314132

R9-DP = 11.22889

90° - 11.22889° = A'

90° - 11.22889° = 78.77708°

 sin A' x cos Vertical Rotate Angle = a

 sin 78.77708° x 0.314132 = 0.3081257

180° -  78.77708° - 16.86990° = B'

180° -  78.77708° - 16.86990° = 84.35302°

a ÷ sin B'  = b

a ÷ sin 84.35302°  = 0.30962837

 0.3081257 ÷ sin 84.35302°  = 0.30962837

 a ÷ sin B'  = b

sin 71.69168 = c

sin 71.69168 = 0.9493798

arctan(b ÷ c) = 18.063089°

arctan(0.30962837 ÷ 0.9493798) = 18.063089°

Rafter Upper Claw Angle = (90° - 43.219179°) - 18.063089° = 28.71824°