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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.
Taking the function 202 + x - 2, we write
y = x + x - 2.
In order to see where the function lies between y=- 2 and y=-2, we let x =
- 7. We find that when x=- - }, y=-21. Similarly if we give to x other values between 0 and — 1, we shall find that y in every case lies between O 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 + b3 + c are parabolas.
Graphs of functions of the form x2 + y2 = r2, or y=+ Vp2 – x?, 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 – boy2 = ca 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.
Exercise 73. Graphs of Quadratic Functions Plot the graphs of the following functions : 1. x? 5. x2
9. 2.2 + 3. 13. E V4 - 3x? 2. 2 x? 6. x2 + x + 1. 10. 3 x2
14. + 15 – 2 x? 3. 4? 7. x2 x + 1. 11. + 14 - 2? 15. +14+ 3x? 4. 2a + 1. 8. x2 + x 1. 12. + V9 - 4c? 16. + 15 + 2?
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:
0° 15° 30° 45° 60° 75° 90° 105° 120° 135° 150° 1650 180°
0.26 .5 1 .87.97 .97 .7 .5 .26 If we represent each unit on the y-axis by }, and each unit on the 2-axis by 30°, the graph is as follows:
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 180°; that it is negative between the values x = -
- 180° and x = 0°, and 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 (384), 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 , the graph is as follows:
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 2 a radians. The period of the sine is therefore said to be 360° or 2 T.
Exercise 74. Graphs of Trigonometric Functions
1. Verify the following plot of the graph of cos a:
2. What is the period of cos x ?
4. What is the period of tan x ?
6. What is the period of cot x?
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 csc u changes.
10. Plot the graphs of sin x and cos on the same paper. What does the figure tell as to the mutual relation of these functions ?
Exercise 75. Miscellaneous Exercise
Find the areas of the triangles in which : 1. a = 25, b= 25, C= 25. 3. a = 74, b= 75, c= 92. 2. a= 25, b = 33}, c = 413. 4. a= 21, b = 3, c=47. 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 w 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 ?
13. How many radians in the angle subtended by an arc 1.62 in. long, the radius being 4.86 in. ? How many degrees ?
Draw the following angles :
20. 3 7 – 9. 15. – 2. 17. -1
19. 2 - 6. 21. 4
1 22. Find four angles whose tangent is
1 23. Find four angles whose cotangent is
3 24. Plot the graphs of sin x and csc x 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 cot x on the same paper. What does the figure tell as to the mutual relation of these functions ?
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
4x – 3 = 7, 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.
Thus sin x =1 is a trigonometric equation. It is true for x = 90° or } , It = 450° or 21, 2 = 810° or 4}+, and so on, with a period of 360° or 2 tt. In general, therefore, the equation sin x =1 is true for x = (2n + })?. It is this general value that is required in solving a general trigonometric equation.
On the other hand, the equation sina x = 1- cosa x 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.
Thus to prove that sin x = 1: csc 3 we have only to substitute for sin x and for csc x. We then see that
Similarly, to prove that tan x = sin x sec x, we may substitute
for sinx, and for sec x. We then have
с =1:-. с
We can often prove a trigonometric identity oy reference to formulas already proved. This was done in proving the identity sin 2 x = 2 sin x cos 2 ($ 101), and in
tana + tany proving tan (x + y) =
1- tan x tany In some cases it may be better to draw a figure and use a geometric proof, as was done in § 90.