Trigonometry.
Trigonometry is that portion of geometry which has for its object the measurement of triangles. When it treats of plane triangles, it is called Plane Trigonometry; and as the engineer will continually meet in his studies of higher mathematics the terms used in plane trigonometry, it is advantageous for him to become familiar with some of the principles and definitions relating to this branch of mathematics.
The circumferences of all circles contain the same number of degrees, but the greater the radius the greater is the absolute measures of a degree. The circumference of a fly wheel or the circumference of the earth have the same number of degrees; yet the same number of degrees in each and every circumference is the measure of precisely the same angle.
The circumference of a circle is supposed to be divided into 360 degrees or divisions, and as the total angularity about the center is equal to four right angles, each right angle contains 90 degrees, or 90°, and half a right angle contains 45°. Each degree is divided into 60 minutes, or 60′; and for the sake of still further minuteness of measurement, each minute is divided into 60″. In a whole circle there are, therefore, 360 × 60 × 60 = 1,296,000 seconds. The annexed diagram, fig. 136, exemplifies the relative positions of the
Sine,
Co-sine,
Versed Sine,
Tangent,
Co-Tangent,
Secant and
Co-secant
of an angle.
Fig. 136.
These may be defined thus:
DEFINITIONS.
1. The Complement of an arc is 90° minus the arc.
2. The Supplement of an arc is 180° minus the arc.
3. The Sine of an angle, or of an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end.
4. The Cosine of an arc is the perpendicular distance from the center of the circle to the sine of the arc; or, it is the same in magnitude as the sine of the complement of the arc.
5. The Tangent of an arc is a line touching the circle in one extremity of the arc, and continued from thence, to meet a line drawn through the center and the other extremity.
6. The Cotangent of an arc is the tangent of the complement of the arc. The Co is but a contraction of the word complement.
7. The Secant of an arc is a line drawn from the center of the circle to the extremity of the tangent.
8. The Cosecant of an arc is the secant of the complement.
9. The Versed Sine of an arc is the distance from the extremity of the arc to the foot of the sine.
For the sake of brevity, these technical terms are contracted thus: for sine A B, we write sin. A B; for cosine A B, we write cos. A B; for tangent A B, we write tan. A B, etc.
Fig. 137.