A recent land survey was conducted on a vacant lot where a commercial building is to be erected. The plans for the future building construction call for a building having a roof supported by two sets of beams. The beams in the front are 8 feet high and the back beams are 6.5 feet high. The distance between the front and back beams is 8 feet. At what angle will the roof lay on the front beam?

6.5ft

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The right triangle formed by the distance between the front and back beams, and the difference between the height of the front and back beams is shown below. From this right triangle, the relationship between the angle  and the two given sides, which is the adjacent (ad=8ft) and the opposite (op=1.5ft) is given by the following trigonometric equation

Solving for q results in

q
op=8ft-6.5 ft

We simply plug-in the values and get the following solution:

If the angle being asked is the angle of inclination downward with respect to the normal of the front beam, then we are talking about q=10.62 degrees but if it is the angle of inclination with respect to the vertical front beam then it is b as shown in the diagram. To solve for beta, since the total angle of the horizontal normal to the front beam and the front beam is 90 degrees then b  can be calculated as shown below

References

Sanchez, L. (1996). Trigonometric Functions . Retrieved from the SOSMath  website: http://www.sosmath.com/trig/Trig2/trig2/trig2.html,  on March 13, 2007

Wesner, T. H. & Mahler, P.H. (1994). The Inverse Since, Cosine and Tangent Functions. In College Algebra & Trigonometry with Applications. (pp. 310-311). IOWA : Wm. C. Brown Communications, Inc.