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3.3. Math Functions

In mathematics, you have probably seen functions like $\sin$ and $\log$, and you have learned to evaluate expressions like $\sin (\pi /2)$ and $\log (1/x)$. First, you evaluate the expression in parentheses, which is called the argument of the function. For example, $\pi /2$ is approximately 1.571, and $1/x$ is 0.1 (if $x$ happens to be 10).

Then you can evaluate the function itself, either by looking it up in a table or by performing various computations. The $\sin$ of 1.571 is 1, and the $\log$ of 0.1 is -1 (assuming that $\log$ indicates the logarithm base 10).

This process can be applied repeatedly to evaluate more complicated expressions like $\log (1/\sin (\pi /2))$. First we evaluate the argument of the innermost function, then evaluate the function, and so on.

C++ provides a set of built-in functions that includes most of the mathematical operations you can think of. The math functions are invoked using a syntax that is similar to mathematical notation:

This program performs calculations using some of the built-in functions from the cmath library.

#include <iostream> #include <cmath> using namespace std; int main () { double result = log (17.0); double angle = 1.5; double height = sin (angle); cout << result << endl; cout << angle << endl; cout << height << endl; return 0; }

The first example sets log to the logarithm of 17, base $e$. There is also a function called log10 that takes logarithms base 10.

The second example finds the sine of the value of the variable angle. C++ assumes that the values you use with sin and the other trigonometric functions (cos, tan) are in radians.

Note

To convert from degrees to radians, you can divide by 360 and multiply by 2 * pi.

If you don’t happen to know $\pi$ to 15 digits, you can calculate it using the acos function. The arccosine (or inverse cosine) of -1 is $\pi$, because the cosine of $\pi$ is -1.

This program also uses built-in functions from the cmath library, specifically the functions that deal with angles. As you can see, we have a line of code that converts the default radians value to degrees.

#include <iostream> #include <cmath> using namespace std; int main () { double pi = acos(-1.0); double degrees = 90; double angle = degrees * 2 * pi / 360.0; cout << pi << endl; cout << degrees << endl; cout << angle << endl; return 0; }

Before you can use any of the math functions, you have to include the math header file. Header files contain information the compiler needs about functions that are defined outside your program. For example, in the “Hello, world!” program we included a header file named iostream using an include statement:

#include <iostream>
using namespace std;

iostream contains information about input and output (I/O) streams, including the object named cout. C++ has a powerful feature called namespaces, that allow you to write your own implementation of cout. But in most cases, we would need to use the standard implementation. To convey this to the compiler, we use the line

using namespace std;

Note

As a rule of the thumb, you should write using namespace std; whenever you use iostream.

Similarly, the math header file contains information about the math functions. You can include it at the beginning of your program along with iostream:

#include <cmath>

Such header files have an initial ‘c’ to signify that these header files have been derived from the C language.

Q-3: Match the statement to its description. This is feedback.
• cmath
• allows the use of functions like log and sin
• iostream
• contains information about input and output streams
• namespace std
• the standard implementation of cout

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Q-5: Multiple Response Select all correct cmath functions.

• cos

• This function computes the cosine of an angle.

• arctan

• The arc tangent function is actually called atan.

• log10

• This function computes the common logarithm.

• pow

• This function raises an expression to a power.

• ln

• The natural log function is actually called log.

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