144. Graph of a Quadratic Function. We shall now consider functions of the second degree in the variable. Such a function is called a quadratic function.

In order to see where the function lies between y = —

2 and y =- 2, we

let x=- - . We find that when x =— -1,y=-24. Similarly if we give to x other values between 0 and — 1, we shall find that y in every case lies between 0 and -2.

Plotting the points and drawing through them a smooth curve, we have the graph as here shown.

This curve is a parabola. All graphs of functions of the form y = ax2 + bx + c are parabolas.

Graphs of functions of the form x2 + y2 = r2, or y = ± √r2 — x2, are circles with their center at 0.

Graphs of functions of the form a2x2 + b2y2 = c2 are ellipses, these becoming circles if a = = b.

Graphs of functions of the form a2x2 - b2y2 = c2 are hyperbolas.

-

There are more general equations of all these conic sections, but these suffice for our present purposes. The graph of every quadratic function in x and y is always a conic section.

145. Graph of the Sine. Since sin x is a function of x, we can plot the graph of sin x. We may represent x, the arc (or angle), in degrees or in radians on the x-axis. Representing it in degrees, as more familiar, we may prepare a table of values as follows:

x = 0° 15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 165° y = 0 .26 .5 .7 .87 .97 1 .97 .87 .7 .5 .26

180°...

0

If we represent each unit on the y-axis by, and each unit on the x-axis by 30°, the graph is as follows:

AA

360° 2709-180 +90°

60 120 180 0 30° 90° 150

$60

The graph shows very clearly that the sine of an angle x is positive between the values x = 0° and x = 180°, and also between the values x =

360° and and

x = 180°; that it is negative between the values x = -180° and x = 0°, also between the values x = 180° and x = 360°. It also shows that the sine changes from positive to negative as the angle increases and passes through - 180° and 180°, and that the sine changes from negative to positive as the angle increases and passes through the values 360°, 0°, and 360°. These facts have been found analytically (§ 84), but they are seen more clearly by studying the graph.

If we use radian measure for the arc (angle), and represent each unit on the x-axis by 0.1 π, the graph is as follows:

AA

157

877

1.670

The nature of the curves is the same, the only difference being that we have used different units of measure on the x-axis, thus elongating the curve in the second figure.

146. Periodicity of Functions. This curve shows graphically what we have already found, that periodically the sine comes back to any given value.

Thus sin x = 1 when x=- 270°, 90°, 450°, ..., returning to this value for increase of the angle by every 360°, or 2π radians. The period of the sine is therefore said to be 360° or 2 π.

Exercise 74. Graphs of Trigonometric Functions

1. Verify the following plot of the graph of cos x:

2. What is the period of cos x ?

3. Verify the following plot of the graph of tan x:

4. What is the period of tan x ?

5. Verify the following plot of the graph of cotx:

L

-860-270 -1809-909 LOP 90% 180° 270°°o 360°

6. What is the period of cot x?

7. Verify the following plot of the graph of secx:

3604210180

+90°

0° 90° 1809 2709 860

8. What is the period of sec x ?

9. Plot the graph of csc x, and state the period. Also state at what values of x the sign of cscx changes.

10. Plot the graphs of sin x and cos x on the same paper. What does the figure tell as to the mutual relation of these functions?

5. Consider the area of a triangle with sides 17.2, 26.4, 43.6. 6. Consider the area of a triangle with sides 26.3, 42.4, 73.9.

7. Two inaccessible points A and B are visible from D, but no other point can be found from which both points are visible. Take some point C from which both A and D can be seen and measure CD, 200 ft.; angle ADC, 89°; and angle ACD, 50° 30'. Then take some point E from which both D and B are visible, and measure DE, 200 ft.; angle BDE, 54° 30'; and angle BED, 88° 30'. At D measure angle ADB, 72° 30'. Compute the distance AB.

8. Show by aid of the table of natural sines that sin x and x agree to four places of decimals for all angles less than 4° 40'.

9. If the values of log x and log sin x agree to five decimal places, find from the table the greatest value x can have.

10. Find four angles whose cosine is the same as the cosine of 175°. 11. Find four angles whose cosine is the same as the cosine of 200°. 12. How many radians in the angle subtended by an arc 7.2 in. long, the radius being 3.6 in.? How many degrees?

24. Plot the graphs of sin x and escx on the same paper. What does the figure tell as to the mutual relation of these functions?

25. Plot the graphs of cos x and sec x on the same paper. What does the figure tell as to the mutual relation of these functions?

26. Plot the graphs of tan x and cotx on the same paper. What does the figure tell as to the mutual relation of these functions?

CHAPTER XI

TRIGONOMETRIC IDENTITIES AND EQUATIONS

147. Equation and Identity. An expression of equality which is true for one or more values of the unknown quantity is called an equation. An expression of equality which is true for all values of the literal quantities is called an identity.

For example, in algebra we may have the equation

which is true only if x = 2.5. Or we may have the identity

(a + b)2 = a2 + 2 ab + b2,

which is true whatever values we may give to a and b.

90° or π,

Thus sin x = 1 is a trigonometric equation. It is true for x = x = 450° or 2 π, x = 810° or 4π, and so on, with a period of 360° or 2π. In general, therefore, the equation sin x = 1 is true for x = (2n+1). It is this general value that is required in solving a general trigonometric equation. On the other hand, the equation sin2x = 1- cos2x is true for all values of x. It is therefore an identity.

=

The symbol is often used instead of to indicate identity, but the sign of equality is very commonly employed unless special emphasis is to be laid upon the fact that the relation is an identity instead of an ordinary equation.

148. How to prove an Identity. A convenient method of proving a trigonometric identity is to substitute the proper ratios for the functions themselves.

с

Thus to prove that sin x = 1: cscx we have only to substitute for sin x and for csc x. We then see that =

a

с

с

a

a

Similarly, to prove that tan x = sin x sec x,

for tanx, for sinx, and for secx. We then have

с

We can often prove a trigonometric identity oy reference to formulas already proved.

This was done in proving the identity sin 2x = 2 sin x cos x (§ 101), and in tantan y proving tan (x + y)

1 tan x tan y

(§ 93).

In some cases it may be better to draw a figure and use a geometric proof, as was done in § 90.