The graphic check which, it will be seen, might have been made use of as a graphic solution, consists in setting one square upon another with the angle of direction and the length of one side determined by the data given. That is, in this problem the protractor is set at 30 degrees and a length of 24 units is taken on the inclined square. The lengths of a and b are then carefully measured by taking a reading of the full inches and reading the remaining fraction to hundredths by means of a sharp pair of dividers and the hundredths scale of the square.

Very many carpenters make use of graphic solutions such as this in determining rafter lengths. A little consideration, however, will show that it is a rather risky method of procedure unless the scale is large and the work scaled small. Graphs serve as easy checks against grave errors upon all kinds of work.

## Example 2

Given A and a. To find B, c, and b. Solution - B = 90 degrees - A.

a a a/c = sin A; c = a c ' sin A b

c = cos A; b = c cos A. c

Substitute the numerical values and check as in Example 1.

## Example 3

Given A and b. To find B, a, and c. Solution - B = 90 degrees - A.

a a/c=sin A; a = c sin A.

c b/c = cos A; c = b/cos A-Substitute the numerical values and check as in Example i.

## Example 4

Given a and c. To find A, B, and b.

a Solution - sin A = - (That is, look in the tables, Appendix II, for the angle which has a sine equal to the result obtained by dividing the numerical value of the side a by the value of the side c.)

B = 90 degrees - A.

b a/c=cos A; b = c cos A.

Substitute numerical values and check as in Example 1.

## Example 5

Given a and b. To find A, B. and c.

a Solution - tan A = a/b

B = 90 degrees - A. . a/c= sin A; c=a/sin A. Substitute numerical values and check as in Examble 1,