Chapter XII: Appendix: I

Natural Trigonometric Functions. Consider the angle _DAE_, Fig. 152. From any point on the line AD drop a line perpendicular to the side _AE_ forming the right triangle ABC. Let _a_ represent the value or length of the side _BC_; let _b_ represent the value of the side _AC_; let _c_ represent the value of the side _AB_. The ratio of the side _a_ to the side _c_ is called the sine of the angle _A_. More concisely stated, _a/c_ = sin _A_. The sine of an angle is the ratio of its opposite side to its hypotenuse, or opposite side over hypotenuse = sine of angle _A_ = sin _A_. In a similar manner:

_b_ adjacent side
--- = --------------- = cosine of angle _A_ = cos _A_.
_c_ hypotenuse

_a_ opposite side
--- = --------------- = tangent of angle _A_ = tan _A_.
_b_ adjacent side

_b_ adjacent side
--- = ---------------- = cotangent of angle _A_ = cot _A_.
_a_ opposite side

_c_ hypotenuse
--- = ---------------- = secant of angle _A_ = sec _A_.
_b_ adjacent side

_c_ hypotenuse
--- = --------------- = cosecant of angle _A_ = csc _A_.
_a_ opposite side

These ratios are known as natural functions of the angle because their values change with every change in the value of the angle.

The lengthening of the sides of the angle should not be mistaken for a change in the value of the angle. Draw to scale very carefully any angle and drop lines from any two points, as at _B_ and _B′_, Fig. 152, which shall be perpendicular to the base line. Measure the sides of the triangles so formed and express their ratios as functions of the angle _A_. Comparing like functions of large and small triangle it will be seen that once an angle is known in degrees, its sine, cosine, etc., are determined irrespective of the length of sides. And, vice versa, if we know the functional values or ratios of certain sides of the right triangle formed about an angle, we have determined the value of the angle in degrees. The Table of Natural Trigonometric Functions, Appendix II, is nothing more than a compilation of these various ratios carefully figured out and placed in the form of a table to assist in the easy solution of problems having to do with the finding of certain parts of a triangle when other parts are given.

With a protractor, measure the angle A of the triangle whose sides were just measured, and compare the ratios of the sides or the functional values with those given in the Table, Appendix III, for the same angle. The larger the scale of the drawing, the greater the accuracy. By making use of the hundredths scale of the framing square together with a finely pointed pair of dividers, variation in values should not be great.

=Solutions of Right Triangles.=--By the solution of right triangles is meant the finding of unknown sides or angles when values of other sides and angles are known.

_Example 1._--Given _A_ = 30 degrees, _c_ = 24;
Find _B, a, b_.

_Solution_--_B_ = 90-30 = 60 degrees. (The sum of the angles
of a triangle equals 180 degrees. _C_ = 90 degrees.)

(1) _a_/_c_ = sin _A_; whence, _a_ = _c_ sin _A_.
(_a_ = _c_ times sine _A_.)

(2) _b_/_c_ = cos _A_; whence _b_ = _c_ cos _A_.

From the Tables, Appendix II, sin of _A_, or 30 degrees, = .5. Substituting numerical values in (1), _a_ = 12.

Again, from Tables, cos _A_, or 30 degrees, = .866. Substituting numerical values in (2), _b_ = 20.784.

Arith. check _c²_ = _a²_ + _b²_; 24² = 12² + 20.78²; 576 = 144 + 431.8; 576 = 576.

Graphic check.=--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_/_c_ = sin _A_; _c_ = _a_/sin _A_

_b_/_c_ = cos _A_; _b_ = _c_ cos _A_.

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_/_c_ = sin _A_; _a_ = _c_ × sin _A_.

_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_.

_Solution_--sin _A_ = _a_ / _c_ (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_/_c_ = cos _A_; _b_ = _c_ cos _A_.

Substitute numerical values and check as in Example i.

_Example 5._--Given _a_ and _b_. To find _A_, _B_. and _c_.

_Solution_--tan _A_ = _a_ / _b_

_B_ = 90 degrees _A_.

_a_/_c_ = sin _A_; _c_ = _a_ / sin _A_

Substitute numerical values and check as in Example 1.