Fundamental Hip-Valley Roof Ratios

Hip-Valley Pitch Angle

arccos (cos Common Rafter Pitch Angle / cos Backing Angle)
arccos (sin Jack Rafter Side Cut Angle / cos Plan Angle)
arccos (cos Sheathing Angle / cos Plan Angle)
arcsin (sin Common Rafter Pitch Angle × cos Jack Rafter Side Cut Angle)
arcsin (sin Common Rafter Pitch Angle × sin Sheathing Angle)
arcsin (tan Plan Angle × tan Backing Angle)
arctan (sin Backing Angle / tan Jack Rafter Side Cut Angle)
arctan (sin Backing Angle × tan Sheathing Angle)
arctan (tan Common Rafter Pitch Angle × sin Plan Angle)

Backing Angle

arccos (sin Plan Angle / cos Jack Rafter Side Cut Angle)
arccos (sin Plan Angle / sin Sheathing Angle)
arccos (cos Common Rafter Pitch Angle / cos Hip-Valley Pitch Angle)
arcsin (sin Common Rafter Pitch Angle × cos Plan Angle)
arcsin (tan Jack Rafter Side Cut Angle × tan Hip-Valley Pitch Angle)
arcsin (tan Hip-Valley Pitch Angle / tan Sheathing Angle)
arctan (tan Common Rafter Pitch Angle × sin Jack Rafter Side Cut Angle)
arctan (tan Common Rafter Pitch Angle × cos Sheathing Angle)
arctan (sin Hip-Valley Pitch Angle / tan Plan Angle)

Jack Rafter Side Cut Angle

arccos (sin Plan Angle / cos Backing Angle)
arccos (sin Hip-Valley Pitch Angle / sin Common Rafter Pitch Angle)
arcsin (cos Plan Angle × cos Hip-Valley Pitch Angle)
arcsin (tan Backing Angle / tan Common Rafter Pitch Angle)
arctan (cos Common Rafter Pitch Angle / tan Plan Angle)
arctan (sin Backing Angle / tan Hip-Valley Pitch Angle)

Sheathing Angle

arccos (cos Plan Angle × cos Hip-Valley Pitch Angle)
arccos (tan Backing Angle / tan Common Rafter Pitch Angle)
arcsin (sin Plan Angle / cos Backing Angle)
arcsin (sin Hip-Valley Pitch Angle / sin Common Rafter Pitch Angle)
arctan (tan Plan Angle / cos Common Rafter Pitch Angle)
arctan (tan Hip-Valley Pitch Angle / sin Backing Angle)

Common Rafter Pitch Angle

arccos (cos Hip-Valley Pitch Angle × cos Backing Angle)
arccos (tan Jack Rafter Side Cut Angle × tan Plan Angle)
arccos (tan Plan Angle / tan Sheathing Angle)
arcsin (sin Hip-Valley Pitch Angle / cos Jack Rafter Side Cut Angle)
arcsin (sin Hip-Valley Pitch Angle / sin Sheathing Angle)
arcsin (sin Backing Angle / cos Plan Angle)
arctan (tan Hip-Valley Pitch Angle / sin Plan Angle)
arctan (tan Backing Angle / sin Jack Rafter Side Cut Angle)
arctan (tan Backing Angle / cos Sheathing Angle)

Plan Angle

arccos (sin Jack Rafter Side Cut Angle / cos Hip-Valley Pitch Angle)
arccos (cos Sheathing Angle / cos Hip-Valley Pitch Angle)
arccos (sin Backing Angle / sin Common Rafter Pitch Angle)
arcsin (tan Hip-Valley Pitch Angle / tan Common Rafter Pitch Angle)
arcsin (cos Backing Angle × cos Jack Rafter Side Cut Angle)
arcsin (cos Backing Angle × sin Sheathing Angle)
arctan (sin Hip-Valley Pitch Angle / tan Backing Angle)
arctan (cos Common Rafter Pitch Angle / tan Jack Rafter Side Cut Angle)
arctan (cos Common Rafter Pitch Angle × tan Sheathing Angle)

Joe Bartok
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