142 Chapter 4   Mathematical Functions, Characters, and Strings

4.1 Introduction
The focus of this chapter is to introduce mathematical functions, characters, string
objects, and use them to develop programs.
Key
Point The preceding chapters introduced fundamental programming techniques and taught you how
to write simple programs to solve basic problems using selection statements. This chapter
introduces methods for performing common mathematical operations. You will learn how to
create custom methods in Chapter 6.
Suppose you need to estimate the area enclosed by four cities, given the GPS locations (lati-
problem tude and longitude) of these cities, as shown in the following diagram. How would you write
a program to solve this problem? You will be able to write such a program in this chapter.
Charlotte (35.2270869, –80.8431267)


Atlanta
(33.7489954, –84.3879824)
Savannah (32.0835407, –81.0998342)




Orlando (28.5383355, –81.3792365)

Because strings are frequently used in programming, it is beneficial to introduce strings early
so that you can begin to use them to develop useful programs. This chapter also gives a brief
introduction to string objects; you will learn more on objects and strings in Chapters 9 and 10.

4.2 Common Mathematical Functions
Java provides many useful methods in the Math class for performing common math-
Key ematical functions.
Point
A method is a group of statements that performs a specific task. You have already used the
pow(a, b) method to compute ab in Section 2.9.4, Exponent Operations and the random()
method for generating a random number in Section 3.7. This section introduces other useful
methods in the Math class. They can be categorized as trigonometric methods, exponent meth-
ods, and service methods. Service methods include the rounding, min, max, absolute, and ran-
dom methods. In addition to methods, the Math class provides two useful double constants,
PI and E (the base of natural logarithms). You can use these constants as Math.PI and Math.E
in any program.

4.2.1 Trigonometric Methods
VideoNote The Math class contains the following methods as listed in Table 4.1 for performing
Introduce Math functions t­rigonometric functions:
Table 4.1 Trigonometric Methods in the Math Class
Method Description
sin(radians) Returns the trigonometric sine of an angle in radians.
cos(radians) Returns the trigonometric cosine of an angle in radians.
tan(radians) Returns the trigonometric tangent of an angle in radians.
toRadians(degree) Returns the angle in radians for the angle in degrees.
toDegrees(radians) Returns the angle in degrees for the angle in radians.
asin(a) Returns the angle in radians for the inverse of sine.
acos(a) Returns the angle in radians for the inverse of cosine.
atan(a) Returns the angle in radians for the inverse of tangent.
 4.2 Common Mathematical Functions 143
The parameter for sin, cos, and tan is an angle in radians. The return value for asin and
atan is an angle in radians in the range between -p/2 and p/2, and for acos is between 0
and p. One degree is equal to p/180 in radians, 90 degrees is equal to p/2 in radians, and 30
degrees is equal to p/6 in radians.
For example,
Math.toDegrees(Math.PI / 2) returns 90.0
Math.toRadians(30) returns 0.5236 (same as π/6)
Math.sin(0) returns 0.0
Math.sin(Math.toRadians(270)) returns −1.0
Math.sin(Math.PI / 6) returns 0.5
Math.sin(Math.PI / 2) returns 1.0
Math.cos(0) returns 1.0
Math.cos(Math.PI / 6) returns 0.866
Math.cos(Math.PI / 2) returns 0
Math.asin(0.5) returns 0.523598333 (same as π/6)
Math.acos(0.5) returns 1.0472 (same as π/3)
Math.atan(1.0) returns 0.785398 (same as π/4)



4.2.2 Exponent Methods
There are five methods related to exponents in the Math class as listed in Table 4.2.

Table 4.2 Exponent Methods in the Math Class
Method Description
exp(x) Returns e raised to power of x (ex).
log(x) Returns the natural logarithm of x (ln(x) = loge(x)).
log10(x) Returns the base 10 logarithm of x (log10(x)).
pow(a, b) Returns a raised to the power of b (ab).
sqrt(x) Returns the square root of x ( 2x) for x 7 = 0.


For example,
e3.5 is Math.exp(3.5), which returns 33.11545
ln(3.5) is Math.log(3.5), which returns 1.25276
log10 (3.5) is Math.log10(3.5), which returns 0.544
23 is Math.pow(2, 3), which returns 8.0
32 is Math.pow(3, 2), which returns 9.0
4.52.5 is Math.pow(4.5, 2.5), which returns 42.9567
24 is Math.sqrt(4), which returns 2.0
210.5 is Math.sqrt(10.5), which returns 3.24



4.2.3 The Rounding Methods
The Math class contains four rounding methods as listed in Table 4.3.

Table 4.3 Rounding Methods in the Math Class
Method Description
ceil(x) x is rounded up to its nearest integer. This integer is returned as a double value.
floor(x) x is rounded down to its nearest integer. This integer is returned as a double value.
rint(x) x is rounded to its nearest integer. If x is equally close to two integers, the even one is returned as a double value.
round(x) Returns (int)Math.floor(x + 0.5) if x is a float and returns (long)Math.floor(x + 0.5) if x is a double.
144 Chapter 4   Mathematical Functions, Characters, and Strings
For example,
Math.ceil(2.1) returns 3.0
Math.ceil(2.0) returns 2.0
Math.ceil(−2.0) returns −2.0
Math.ceil(−2.1) returns −2.0
Math.floor(2.1) returns 2.0
Math.floor(2.0) returns 2.0
Math.floor(−2.0) returns −2.0
Math.floor(−2.1) returns −3.0
Math.rint(2.1) returns 2.0
Math.rint(−2.0) returns −2.0
Math.rint(−2.1) returns −2.0
Math.rint(2.5) returns 2.0
Math.rint(4.5) returns 4.0
Math.rint(−2.5) returns −2.0
Math.round(2.6f) returns 3 // Returns int
Math.round(2.0) returns 2 // Returns long
Math.round(−2.0f) returns −2 // Returns int
Math.round(−2.6) returns −3 // Returns long
Math.round(−2.4) returns −2 // Returns long



4.2.4 The min, max, and abs Methods
The min and max methods return the minimum and maximum numbers of two numbers (int,
long, float, or double). For example, max(4.4, 5.0) returns 5.0, and min(3, 2)
returns 2.
The abs method returns the absolute value of the number (int, long, float, or double).
For example,
Math.max(2, 3) returns 3
Math.min(2.5, 4.6) returns 2.5
Math.max(Math.max(2.5, 4.6), Math.min(3, 5.6)) returns 4.6
Math.abs(−2) returns 2
Math.abs(−2.1) returns 2.1



4.2.5 The random Method
You used the random() method in the preceding chapter. This method generates a random
double value greater than or equal to 0.0 and less than 1.0 (0 <= Math.random() < 1.0).
You can use it to write a simple expression to generate random numbers in any range. For
example,


Returns a random integer
(int)(Math.random() * 10)
between 0 and 9.
Returns a random integer
50 + (int)(Math.random() * 50)
between 50 and 99.


In general,


Returns a random number between a
a + Math.random() * b
and a + b, excluding a + b.
 4.2 Common Mathematical Functio...