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De ne the function Tan x to be the restriction of tan x to the interval ( /2, /2) Observe that the tangent function is unde ned at the endpoints of this interval Since d Tan x = sec2 x dx we see that Tan x is increasing, hence it is one-to-one (Fig 619) Also Tan takes arbitrarily large positive values when x is near to, but less than, /2 And Tan takes negative values that are arbitrarily large in absolute value when x is near to, but greater than, /2 Therefore Tan takes all real values Since Tan : ( /2, /2) ( , ) is one-to-one and onto, the inverse function Tan 1 : ( , ) ( /2, /2) exists The graph of this inverse function is shown in Fig 620 It is obtained by the usual procedure of re ecting in the line y = x EXAMPLE 638

Part II:

// This creates a semaphore that allows up to two // permits to be granted and that initially has // two permits available static Semaphore sem = new Semaphore(2, 2); public MyThread(string name) { Thrd = new Thread(thisRun); ThrdName = name; ThrdStart(); } // Entry point of thread void Run() { ConsoleWriteLine(ThrdName + " is waiting for a permit"); semWaitOne(); ConsoleWriteLine(ThrdName + " acquires a permit"); for(char ch='A'; ch < 'D'; ch++) { ConsoleWriteLine(ThrdName + " : " + ch + " "); ThreadSleep(500); } ConsoleWriteLine(ThrdName + " releases a permit"); // Release the semaphore semRelease(); } }

class SemaphoreDemo { static void Main() { // Construct MyThread mt1 MyThread mt2 MyThread mt3 three = new = new = new threads MyThread("Thread #1"); MyThread("Thread #2"); MyThread("Thread #3");

Calculate Tan 1 1, Tan 1 1/ 3,

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mt1ThrdJoin(); mt2ThrdJoin(); mt3ThrdJoin(); } }

MyThread declares the semaphore sem, as shown here:

static Semaphore sem = new Semaphore(2, 2);

This creates a semaphore that can grant up to two permits and that initially has both permits available

Tan 1 ( 3)

23:

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In MyThreadRun( ), notice that execution cannot continue until a permit is granted by the semaphore, sem If no permits are available, then execution of that thread suspends When a permit does become available, execution resumes and the thread can run In Main( ), three MyThread threads are created However, only the first two get to execute The third must wait until one of the other threads terminates The output, shown here, verifies this (The actual output you see may vary slightly)

Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread Thread #1 #1 #1 #2 #2 #2 #3 #1 #2 #1 #2 #1 #3 #3 #2 #3 #3 #3 is waiting acquires a : A is waiting acquires a : A is waiting : B : B : C : C releases a acquires a : A releases a : B : C releases a for a permit permit for a permit permit for a permit

SOLUTION We have Tan 1 1 = , 4 1 Tan 1/ 3 = , 6 1 Tan ( 3) = 3

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