Functions of 150°. From the figure and definitions it is

PROBLEM. Find the values of the trigonometric functions of 210°, 225°, 240°, 300°, 315°, and 330°.

24. Signs of the trigonometric ratios. The signs of the trigonometric ratios of any angle depend upon the signs of the ordinate, the abscissa, and the distance of any point on its terminal line. As the terminal line passes from one quadrant to another, there is always a change of sign in either the abscissa or the ordinate of any point on that line, but the distance remains positive. When a coordinate changes its sign, every trigonometric ratio dependent upon it must also change its sign.

The following table is constructed by taking account of the signs of the abscissa x and of the ordinate y, and remembering that the distance r is always positive. It gives the sign of each trigonometric ratio of an angle terminating in any quadrant.

25. Theorem.

For every given angle there is one and only

one value of each trigonometric function.

ferent points on the terminal line.

Let P and P2 be any two points on the terminal line. Then by definition

AP2.

or sin α =

OP

But since the right triangles OАР1 and О.AP2 are similar,

and have their corresponding sides in the same direction, it follows that

AP_AP2.

=

OP OP2

Hence the value of sin a is independent of the position of the point chosen on the terminal line, but depends solely upon the position of the terminal line, i.e. upon the angle. The above theorem may be stated as follows: The trigonometric functions are single-valued functions of the angle.

26. Theorem. Every given value of a trigonometric function determines an unlimited or infinite number of positive and negative angles, among which there are in general two positive angles less than 360°.

The theorem is demonstrated for a given tangent. A similar method is applicable to the remaining functions.

and the origin determine the terminal line of the angle a1, whose tangent is

m n

Likewise the point P2 is located by using - m and n as ordinate and abscissa respectively. Drawing the terminal line OP2, a second angle a is found, which also has the given tangent.

The angles a1 and α2, are evidently less than 360°, and have the given tangent There is an unlimited num

m
n

ber of positive and negative angles coterminal with a, and α2, all of which have the given tangent. Hence the theorem.

7. sin a is positive and cos a is negative. 8. tan a is positive and cos a is negative. 9. cosec a is negative and cos a is negative. 10. tan a is negative and sin a is positive. 11. cos a is negative and sin a is negative.

Give the signs of the trigonometric functions of the fol

Find the negative angles, numerically less than 360°, that are coterminal with the following angles:

26. Construct the positive angles, less than 360°, for which the sine is equal to %, and find the values of the other functions of both angles.

27. Construct the positive angles, less than 360°, for which the cosine is g, and find the values of the other functions of both angles.

Find the values of the functions of all angles less than 360° determined by

35. Given tan α=— -4, find the value of

36. Given sec α = 6, find the value of

sin a cos a
cot a

sin2 α + cos2 α
cos2 α

37. Given sin α = .3, find the value of tan a sec a cos α.

38. Given csc α = 8 and

sin a cos ẞ+ cos α sin ß.

39. Given tan α = and cos ẞ=-√5, a terminating in the first quadrant and ß in the second, show that the angle between the terminal lines of a and ẞ is a right angle.