7. Conversely, any angle expressed in circular measure may be reduced to degrees by multiplying by 180° and dividing by ; or, more briefly, by substituting 180° for π.

8. In the circular method such expressions may occur as

These refer to the unit of circular measure; that is, the

2

angle means an angle whose subtending arc is two-thirds

3

of the radius.

The angle 1, that is, the angle whose subtending arc is equal to the radius, or the unit of circular measure, if reduced to degrees by the rule of Art. 7, gives

The rule of Art. 7 may then be modified as follows:

Any angle expressed in circular measure may be reduced to degrees by multiplying by 57.2957795°...

II. THE TRIGONOMETRIC FUNCTIONS.

FUNCTIONS OF ACUTE ANGLES.

10. Let BAC be any acute angle.

B

C.

From any point in either side, as B, draw BC perpendic ular to the other side, forming a right triangle ABC.

We then have the following definitions, applicable to either of the acute angles A and B:

In any right triangle,

The SINE

The COSINE

The TANGENT

of either acute angle is the ratio of the opposite side to the hypotenuse.

is the ratio of the adjacent side to the hypot

enuse.

is the ratio of the opposite side to the adjacent side.

The COTANGENT is the ratio of the adjacent side to the oppo

The SECANT

The COSECANT

site side.

is the ratio of the hypotenuse to the adjacent side.

is the ratio of the hypotenuse to the opposite side.

That is, denoting the sides BC, CA, and AB by a, b, and c, and employing the usual abbreviations,

11. The following definitions are also used:

The versed sine of an angle is equal to unity minus the cosine of the angle.

The coversed sine is equal to unity minus the sine.

12. The eight ratios defined in Arts. 10 and 11 are called Trigonometric Functions, or Trigonometric Ratios, of the angle.

It is important to observe that their values depend solely on the magnitude of the angle, and are entirely independent of the lengths of the sides of the right triangle which contains it.

B

B'

A

C'

Thus, let B and B' be any two points in the side AD of the angle DAE, and draw BC and B'C' perpendicular to AE.

But the right triangles ABC and AB'C' are similar, since they have the angle A common; and therefore, by Geometry,

Thus the two values obtained for sin A are seen to be equal.

13. We obtain from (1) and (2), Art. 10,

That is, in any right triangle, either side about the right angle is equal to the hypotenuse multiplied by the sine of the opposite angle, or by the cosine of the adjacent angle.

Since the angles A and B are complements of each other, the above results may be expressed as follows:

The sine, tangent, secant, and versed sine of an acute angle are respectively the cosine, cotangent, cosecant, and coversed sine of the complement of the angle.

It is from this circumstance that the names co-sine, cotangent, etc., were derived.

15. The Pythagorean Theorem affords a simple method for finding the values of the remaining seven functions of an acute angle, when the value of any one is given.

1. Given cot A= 2; required the values of the remaining functions of A.

Then since the cotangent is the ratio of the adjacent side to the opposite side, we may regard our value as having been taken from a right triangle ABC, having its side AC adjacent to the angle A equal to 2 units, and its side BC opposite to A equal to 1 unit.

But by Geometry, we have

AB=√AC2 + BC2 = √4 + 1 = √5.

Whence by definition,