If the height of a gable end of a roof is 22.5 feet and the rafters are 30 feet 8 inches long, at what angle do the rafters slope, and how wide is the gable end at the base?
A gable end ABD of a roof is an isosceles triangle with the base being the width of the house, and the two equal sloping sides the rafters at the end of the roof. If you drop a perpendicular from the apex B of the triangle, you’ll get two congruent right triangles, ABC and DBC. Since you know two sides of the right triangle ABC, you can compute the third by using the Pythagorean theorem. You can use sines to determine the angle of slope, since sin A = BC/AB = 22.5'/30'8" = 0.7337. To find the angle A, you’ll need what’s called the arcsine of 0.7337. The arcsine function is inverse to the sine function, and your calculator can compute them. Usually there’s a button on the calculator labeled “inv” or “arc” that you press before pressing the sin button. Then you’ll have the angle. Your calculator can probably be set to either degree mode or radian mode. If it’s set to degree mode, then you’ll get the angle in degrees; and if it’s set to radian mode, then you’ll get the angle in radians. Always be sure you know which mode your calculator’s set to.
What is the definition of a limit?
We say that a limit of f(x) is L as x approaches a and write this as the limit as x approaches a for f(x) is equal to L provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.
What is the definition of an inverse function?
Let f and g be two functions such that f(g(x)) = x, for every x in the domain of g g(f(x) = x, for every x in the domain of f, then the function g is said to be the inverse of the function f and is denoted f^-1. The domain of f is equal to the range of f^-1 and the range of f is the domain of f^-1.