1. Discuss the periodicity, zeros, values of @ for which the function becomes infinite, sign, maxima and minima, changes and symmetry of cos 0 and tan 0.

2. Sketch on the same axes the graphs of the pairs of functions following, and discuss the second function with respect to the properties listed in Exercise 1.

(a) sin and csc 0. (b) cos and sec 0. (c) tan and cot 0. 3. Construct the graphs of the six trigonometric functions on the same

axes.

4. What properties of the functions can be inferred from graphs on the same axes of

(a) sin and cos 0? (b) tan 0 and cot 0? (c) sec and csc 0?

5. Describe the motion of a particle on a straight line if its distance s from a fixed point on the line at any time t is given by s sin t. Does such a motion approximate any motion occurring in nature?

6. On the same axes sketch the graphs of the functions:

=

8. By the addition of ordinates (see Exercise 4, page 44) construct the graph of

(a) sin x + cos x.

(b) 2 sin x

COS X.

(e) sin x + 3 cos x.

62. Functions of Complementary Angles. Construct an acute angle and its complement 90° - with their initial lines coinciding with the positive part of the x-axis. On their terminal lines take OP = O'P', so that r = r'. Then P and P' are symmetrical with respect to OA,

the bisector of the first quadrant y

(why?), and hence x = y' and y = x'.

P

The cosine, cotangent and cosecant of an angle are so named because they are respectively the sine, tangent, and secant of the complementary angle, as is shown by these relations. The former functions are called the cofunctions of the latter, respectively, and vice versa. With this terminology, the six relations (1) to (5) may be stated as the

Theorem. The functions of any acute angle are equal respectively to the cofunctions of the complementary angle.

A method of extending the proofs of these relations for any value of 0, not necessarily acute, will be given in Section 68.

63. Tables of Trigonometric Functions. As a consequence of the theorem in the preceding section, tables of values of the trigonometric functions may be printed in very compact form. Since cos (90° — 0) sin 0, a table of sines of any set of angles is also a table of cosines of the complementary angles. The complements of 0°, 1o, 2o, . . ., 88°, 89°, 90° are respectively 90°, 89°, 88°, 2°, 1°, 0°. Hence:

=

1. The table of sines on pages 8 and 9 of Huntington's Tables is also a table of cosines if read backward.

2. The sines and cosines of angles from 0° to 45° in the Condensed Table on the inside of the back cover of the Tables are, if read upward, the cosines and sines respectively of the angles from 45° to 90°.

In like manner, tan 0 and cot ◊ may be given in one table, and so also may sec 0 and csc 0.

Thus the theorem on functions of complementary angles makes it practicable to reduce by one-half the space devoted to a table of trigonometric functions.

Fractional parts of a degree, in Huntington's Tables, are given in tenths and hundredths instead of in minutes and seconds. One of the merits of this decimal method of subdivision is that the process of interpolation, in finding a function of 0, is identical with that used earlier in the other tables (page 121).

B

0.4321

F

sin

I

0.4332

H

D

0.4337

a function of 0 is given, is illustrated in the examples following.

EXAMPLE 1. Find 0 if sin0 = 0.4321.

Searching through the body of the table of sines, we find that 0.4321 is given in the table, and reference to the margin shows that 0 = 25°.6. EXAMPLE 2. Find 0 if sin

=

0.4332.

A search in the body of the table of sines shows that 0.4332 lies between 0.4321 and 0.4337, which are the sines of 25°.6 and 25°.7. Fig. 98 shows the graph of sin 0 between these angles, on the assumption that it is straight (compare the assumption on page 122). To find 0, graphically, is to find the value of 0 at E if EF = 0.4332.

The slope of the graph is the difference of the ordinates of B and D, HD = 0.0016, divided by the difference of the abscissas, AC 0.1 (Definition, page 50). It is also equal to the difference of the ordinates of F and D, ID = 0.0005, divided by the difference of the abscissas, EC.

The arithmetical processes used in interpolating to find an angle of which a function is given are:

Find the two successive numbers in the proper table between which the given number lies.

Find the difference between the given number and that one of these two numbers to which it is nearer, and divide this difference by the tabular difference.

Apply the result as a correction to the last digit of the angle whose function was used in getting the difference above, in such a way that the result lies between the two angles corresponding to the two numbers in the table.

Searching through the body of the table, we find that 0.4815 lies between 0.4818 and 0.4802, the cosines of 61°.2 and 61°.3. It is nearer the former, the difference being 3, neglecting the decimal point, while the tabular difference is 16. The correction to be applied to 61°.2 is therefore of one-tenth of a degree, so that 0 61°.2% 61°.22.

=

=

In using the table of tenths of the tabular difference, in the margin, find the difference 3 as above. Then look for 3 in the margin. It lies in the column headed 2. Hence 3 is 2-tenths of the tabular difference, and the digit 2 is annexed to 61°.2, giving 61°.22.

The use of the tables for other than acute angles will be considered in Section 69.

EXERCISES

1. Find the cosine, cotangent, and cosecant of 72°.43. Find the sine, tangent, and secant of 17°.57. Compare the two sets of results.

2. Find the six functions of 73°.26, and verify the fact that the last three functions are the reciprocals of the first three in the reverse order. 3. Find the acute value of 0, illustrating the interpolation graphically,

if

4. The arithmetical operations in this exercise should be performed mentally. Find the acute value of 0 such that

5. Construct the line through the origin whose slope is 2, and find the angle between the line and the x-axis.

6. Construct a table of values of 0 and sin 0 for values of @ taken every 15° from 0° to 90°, expressing 0 in radians decimally instead of in terms of π (see Tables, page 32), and giving the values of 0 and sin 0 to two decimal places. Construct the graph as accurately as possible from this table, using a large scale. On the same axes draw the graph of 0. What approximate value of sin 0 for very small angles is suggested by these graphs? Using the Condensed Tables on the inside of the back cover of the Tables, determine for how large an angle this approximation is correct to three decimal places; to four decimal places. What is the limit of sin 0/0 as approaches zero?

7. Solve Exercise 6 replacing sin 0 by tan 0.

8. Using the properties of the functions sin 0, cos 0 and tan 0 suggested by the symmetry of their graphs, find the sine, cosine, and tangent of the negative angles, - 24°.32, - 48°.27, — 68°.46.

y

B

a

64. Solution of Right Triangles. It is customary to denote the magnitude of the angles of a triangle ABC by A, B, C and the lengths of the sides opposite by a, b, c respectively. In a right triangle, the right angle is usually denoted by C, and the hypotenuse by c. A right triangle ABC may be placed with reference to coördinate axes so that A is at the origin, AC lies along

Ο Α

b

FIG. 99.

С х