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The square root of a given positive number is the number whose square is the given number. The square root of 9 is 3 because the square of 3 is 9. We can think of the square root function as a black box. When you put in a 9, out comes a 3. When the number 2 is input, the number 1.41421 is output. This function has the same input-process-output nature that complete programs have. However, the processing step is hidden: we do not need to know what the function does to 2 to produce 1.41421. All we need to know is that the output 1.41421 does have the square root property: its square is the input 2. Here is a simple program that uses the predefined square root function: #include <iostream.h> #include <math.h> // Test-driver for the sqrt function: main0 { for (int i = 0; i < 6; i++) tout C-C "\t" -CC i C-C "\t" C-C sqrt(i) << endl; >

Fig. 2-24

This program prints the square roots of the numbers 0 through 5. Each time the expression sqr t ( i > is evaluated in the for loop, the sqr t function is executed. Its actual code is hidden away within the Standard C Library. In using it, we may confidently assume that the expression sqrt ( i ) will be replaced by the actual square root of whatever value i has at that moment. Notice the directive # inc lude <math . h> on the second line. This is necessary for the compiler to find the definition of the sqr t function. It tells the compiler that the function is declared in the <math. h> header file.

A function like sqr t ( ) is executed by using its name as a variable in a statement, like this:

This is a combination of the following two principles: 1. Supplements of the same angle are congruent. Thus, j a > j b in Fig. 2-25 and each is the supplement of j x. 2. Supplements of congruent angles are congruent. Thus, j c > j d in Fig. 2-26 and their supplements are the congruent angles x and y.

y = sqrt(x);

(Notice the negative value assigned to x.)

Fig. 2-25

This is called invoking or calling the function. Thus in Example 4.1, the code sqrt ( i ) caZZs the sqrt function. The expression i in the parentheses is called the argument or actual parameter of the function call, and we say that it is passed by vaZue to the function. So when i is 3, the value 3 is passed to the sqr t function by the call sqr t ( i ) . This process is illustrated by the following diagram:

PRINCIPLE 5:

1.73205 /

Fig. 2-26

sqrt()

The variable i is declared in main ( ) . During the fourth iteration of the for loop, its value is 3. That value is passed to the sqrt ( ) function which then returns the value 1.73205.

5

Here is another program that uses the <math . h> header file. Its purpose is to allow an empirical verification of the standard trigonometric identity sir-h = 2 sinxcosx :

5. (a) Two sides are congruent and i. (b) Opposite sides are congruent. (c) Opposite angles are congruent. (d) AD and BC are congruent and parallel (AD > EF > BC). 6. (a) x 7. (a) x 10. (a) x (f) x 13. (a) x 14. (a) x (c) x 15. (a) m 16. (a) x 17. (a)

#include <iostream.h> #include <math.h> // Program to test trigonometric identity sin 2x = 2 sin x cos x: main0 for (float x = 0; x < 2; x += 0.2) tout << "\t" << x << "\t\t" << sin(2*x) cc 2*sin(x)*cos(x) C-C endl;

When the program is executed, the following error message will be displayed:

1 222;

0 O-0

1 42;

<< "\t"

1 172;

0*389418

0.9 32039 . -. o-6 oi ,0.999574 1 0.909297 f*2 O,Q754G3 1.4 0 t J34988 I..6 . . -0Jl583744 1.8 -0.442521

45 4; (e) x 25, y 25;

0*389428 0.717356 0.932039, 0 * 999574 0,909297 0.675463 01334988 -0,0583744 -0.442521

1 232

y Log(x)

The program prints x in the first column, sin 2x in the second column, and 2 sin x cos x in the third column. For each value of x tested, sin 2x = 2 sin x cos X. Of course, this does not prove the identity. It merely provides convincing empirical evidence of its truth.

12 (since y does not join midpoints, Pr. 3 does not apply);

y = sqrt(2); tout << 2*sin(x)*cos(x);

18. (a) 21; (b) 30; (c) 14; (d) 26