REDUCTION OF FUNCTIONS OF n 90° ± a

44. The formulas to be developed in this section enable us to express any function of any angle in terms of a func tion of an angle differing from the given angle by any multiple of 90°. They may be used to express any function of any angle in terms of a function of an angle less than 90° or less than 45°.

45. Functions of

a in terms of functions of a(n = 0). Let XOP be any positive angle and XOP' a numerically equal negative angle.

Then taking OP' = OP we have, for each figure,

46. Functions of 90° + a in terms of functions of a(n = 1) Let XOP be any positive angle a and XOP' the angle 90° + α.

Then taking OP' OP we have, for each figure,

=

where x, y, and x', y', r' are associated with P and P' respectively.

Similar figures can be constructed for the other quadrants Then in each quadrant

X

- = sec α

=

1).

47. Functions of 90° — a in terms of functions of a (n Let XOP be any positive angle and XOP' the angle

Then taking OP' = OP we have, for each figure,

Similar figures can be constructed for the other quadrants. Then in each quadrant

48. Functions of 180° — a in terms of functions of a (n=2). Let XOP be any positive angle and XOP' the angle 180°

· α.

Then taking OP' OP we have, for each figure,

P' Y

=

:

x' = — x, y' =y, r' = r.

Similar figures can be constructed for the other quadrants.

49. Laws of reduction. The method of the last four articles may be applied to any angle of the form n 90° ± α to obtain formulas of reduction for all positive and negative integral values of n, a being any angle. A complete investigation would show the following laws to be true:

When n is even, any function of n 90° ± a is numerically equal to the same function of a; when n is odd, any function of n 90° ± a is numerically equal to the cofunction of a.

If the function of n 90° ±a is positive, a being considered acute, the members of the equation have like signs; if the function of n 90° ±a is negative, a being considered acute, the members of the equation have unlike signs. The results thus obtained are valid for all positive and negative values of α.

Express as functions of a positive angle less than 90°:

Express each of the trigonometric functions of the following angles as functions of a without using the laws of reduction.

16. 180° + α.

18. 270° + α.

17. 270°

a.

19. 360° -α.