document.write( "Question 624826: Find the exact value of\r
\n" ); document.write( "\n" ); document.write( " Sin(Tan-1 7/24)
\n" ); document.write( " Cos-1(Cos 4Pi/3)
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #393093 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The key to both of these problems is knowing the range of the various inverse trig functions.

\n" ); document.write( "For the first problem, the inverse tan function has a range of 0 to $\"pi\"$ . And since we have been given a positive tan value, 7/24, the angle must be in the range 0 to $\"pi%2F2\"$ , i.e. a first quadrant angle. Since sin is also positive in the first quadrant we will end up with a positive result.

\n" ); document.write( "The only question remaining is: What positive number. Since we're asked for exact values we must not use the Trig buttons on our calculators. To find our solution, we imagine (or draw) a right triangle and pick one of the acute angles. Since tan is opposite over adjacent, make the opposite side 7 and the adjacent side 24. Since sin is opposite over hypotenuse, we need to find the hypotenuse. Use the Pythagorean Theorem to find the hypotenuse:
\n" ); document.write( " $\"c%5E2+=+7%5E2+%2B+24%5E2\"$
\n" ); document.write( "You should find the hypotenuse to be 25. So $\"sin%28tan%5E%28-1%29%287%2F24%29%29+=+7%2F25\"$

\n" ); document.write( "For the second problem one might think that the inverse cos of the cos of $\"4pi%2F3\"$ would be $\"4pi%2F3\"$ ! But this would not be correct because the range of the inverse cos is 0 to $\"pi\"$ and $\"4pi%2F3\"$ is not in this range. So whatever answer we get, it must be between 0 and $\"pi\"$ . $\"4pi%2F3\"$ is an angle which terminates in the 3rd quadrant with a reference angle of $\"4pi%2F3+-+pi+=+pi%2F3\"$ . Since $\"cos%28pi%2F3%29+=+1%2F2\"$ and since cos is negative in the 3rd quadrant, $\"cos%284pi%2F3%29+=+-1%2F2\"$

\n" ); document.write( "Substituting this into our expression we get:
\n" ); document.write( " $\"cos%5E%28-1%29%28-1%2F2%29\"$
\n" ); document.write( "Now we just have to figure out what angle, between 0 and $\"pi\"$ has a cos of -1/2?. Well the reference angle is still $\"pi%2F3\"$ . And in the range 0 to $\"pi\"$ only 2nd quadrant angles have negative cos values. So the angle we are looking for is the second quadrant angle with a reference angle of $\"pi%2F3\"$ :
\n" ); document.write( " $\"pi+-+pi%2F3+=+2pi%2F3\"$
\n" ); document.write( "So $\"cos%5E%28-1%29%28cos%284pi%2F3%29%29+=+2pi%2F3\"$ \n" ); document.write( "

\n" );