III. When A lies in the 3d quadrant, as the reflex angle AOP, MP is negative because measured from M downwards, OM is negative, and OP is positive.

Hence in the third quadrant the sine, cosine, secant, and cosecant, are negative, but the tangent and cotangent are positive.

IV. When A lies in the 4th quadrant, as the reflex angle AOP, MP is negative, OM is positive, and OP is positive.

Hence in the fourth quadrant the sine, tangent, cotangent, and cosecant are negative, but the cosine and secant are positive.

The signs of the different functions are shown in the annexed table.

B

NOTE. It is apparent from this table that the signs of all the functions in any quadrant are known when those of the sine and cosine are known. The tangent and cotangent are + or -, according as the sine and cosine have like or different signs. 19. Changes in the Value of the Sine as the Angle increases from 0° to 360°. Let A denote the angle AOP described by the revolution of OP from its initial position OA through 360°. Then, PM being drawn perpendicular to AA',

P

Α'

M

M

A

M

M

When the angle A is 0°, P coincides with A, and MP is zero; therefore sin 0°

= 0.

As A increases from 0° to 90°, MP increases from zero to OB or OP, and is positive; therefore sin 90° = 1.

Hence in the 1st quadrant sin A is positive, and increases from 0 to 1.

As A increases from 90° to 180°, MP decreases from OP to zero, and is positive; therefore sin 180° = 0.

Hence in the 2d quadrant sin A is positive, and decreases from 1 to 0.

As A increases from 180° to 270°, MP increases from zero to OP, and is negative; therefore sin 270° — — 1.

Hence in the 3d quadrant sin A is negative, and decreases algebraically from 0 to 1.

0.

As A increases from 270° to 360°, MP decreases from OP to zero, and is negative; therefore sin 360° Hence in the 4th quadrant sin A is negative, and increases algebraically from 1 to 0.

20. Changes in the Cosine as the Angle increases from 0° to 360°. — In the figure of Art. 19

When the angle A is 0°, P coincides with A, and OM = OP; therefore cos 0°= 1.

As A increases from 0° to 90°, OM decreases from OP to. zero and is positive; therefore cos 90° = 0.

Hence in the 1st quadrant cos A is positive, and decreases from 1 to 0.

As A increases from 90° to 180°, OM increases from zero to OP, and is negative; therefore cos 180° =- - 1.

Hence in the 2d quadrant cos A is negative, and decreases algebraically from 0 to

-

1.

As A increases from 180° to 270°, OM decreases from OP to zero, and is negative; therefore cos 270° = 0.

Hence in the 3d quadrant cos A is negative, and increases algebraically from 1 to 0.

As A increases from 270° to 360°, OM increases from zero to OP, and is positive; therefore cos 360° = 1.

Hence in the 4th quadrant cos A is positive, and increases from 0 to 1.

21. Changes in the Tangent as the Angle increases from 0° to 360°. — In the figure of Art. 19

As A increases from 0° to 90°, MP increases from zero to OP, and OM decreases from OP to zero, so that on both accounts tan A increases numerically; therefore tan 90° = Hence in the 1st quadrant tan A is positive, and increases from 0 to ∞.

=8.

As A increases from 90° to 180°, MP decreases from OP to zero, and is positive, OM becomes negative and decreases algebraically from zero to 1; therefore tan 180°= 0.

Hence in the 2d quadrant tan A is negative, and increases algebraically from to 0.

When A passes into the 2d quadrant, and is only just greater than 90°, tan A changes from +∞ to

As A increases from 180° to 270°, MP increases from zero to OP, and is negative, OM decreases from OP to zero, and is negative; therefore tan 270° = ∞.

Hence in the 3d quadrant tan A is positive, and increases from 0 to 0.

As A increases from 270° to 360°, MP decreases from OP to zero, and is negative, OM increases from zero to OP, and is positive; therefore tan 360° = 0.

Hence in the 4th quadrant tan A is negative, and increases algebraically from ∞ to 0.

The student is recommended to trace in a manner similar to the above the changes in the other functions, i.e., the cotangent, secant, and cosecant, and to see that his results agree with those given in the following table.

22. Table giving the Changes of the Trigonometric Functions in the Four Quadrants.

NOTE 1. The cosecant, secant, and cotangent of an angle A have the same sign as the sine, cosine, and tangent of A respectively.

The sine and cosine vary from 1 to 1, passing through the value 0. never greater than unity.

They are

The secant and cosecant vary from 1 to -1, passing through the value. They are never numerically less than unity.

The tangent and cotangent are unlimited in value. They have all values from

to +∞.

81

The versed sine and coversed sine vary from 0 to 2, and are always positive. The trigonometrie functions change sign in passing through the values 0 and ∞, and through no other values.

In the 1st quadrant the functions increase, and the cofunctions decrease. NOTE 2. From the results given in the above table, it will be seen that, if the value of a trigonometric function be given, we cannot fix on one angle to which it belongs exclusively.

Thus, if the given value of sin A be, we know since sin A passes through all values from 0 to 1 as A increases from 0° to 90°, that one value of A lies between 0°

and 90°. But since we also know that the value of sin A passes through all values between 1 and 0 as A increases from 90° to 180°, it is evident that there is another value of A between 90° and 180° for which sin A = }.

α

C

23. Relations between the Trigonometric Functions of the Same Angle. Let the radius start from the initial position OA, and revolve P in either direction, to the position OP. Let denote the angle traced out, and let the lengths of the sides PM, MO, OP be denoted by the letters a, b, c.* The following relations are evident from the definitions (Art. 13):

M b

A

Formulæ I., II., III., IV. are very important, and must be remembered.

* a, b, c are numbers, being the number of times the lengths of the sides contain some chosen unit of length.