document.write( "Question 808932: What is the exact value of this expression? arcsin(sin(-3pi/5)) \r
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Algebra.Com's Answer #487323 by KMST(5328) $\"\"$ $\"About$

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THE SHORT ANSWER:
\n" ); document.write( " $\"arcsin%28sin%28-3pi%2F5%29%29=-2pi%2F5\"$ because
\n" ); document.write( " $\"sin%28-3pi%2F5%29%29=sin%28-2pi%2F5%29\"$ and $\"-pi%2F2%3C-2pi%2F5%3Cpi%2F2\"$
\n" ); document.write( "To find the answer you have to look for the angle that
\n" ); document.write( "has the same sine as $\"-3pi%2F5\"$
\n" ); document.write( "and is between $\"-pi%2F2\"$ and $\"pi%2F2\"$
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$\"sin%28-3pi%2F5%29=sin%28AOD%29=y%5BD%5D=y%5BC%5D=sin%28AOC%29=sin%28-2pi%2F5%29\"$
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\n" ); document.write( "THE LONG STORY:
\n" ); document.write( " $\"f%28x%29=sin%28x%29\"$ is a function and its graph looks like this:
\n" ); document.write( " $\"graph%28900%2C100%2C-2%2C18%2C-1.5%2C1.5%2Csin%28x%29%2C1%2C-1%29\"$
\n" ); document.write( "It is a periodic function with period $\"2pi\"$ ,
\n" ); document.write( "meaning that the same value of $\"f%28x%29\"$ is guaranteed to repeat at $\"2pi\"$ intervals.
\n" ); document.write( "The function has a maximum where $\"f%28x%29=1\"$ and it repeats that value $\"2pi\"$ later, at $\"x%2B2pi\"$ , where $\"f%28x%2B2pi%29=1\"$ again.
\n" ); document.write( "For example, $\"sin%28pi%2F2%29=1=sin%28pi%2F2-2pi%29=sin%28pi%2F2%2B2pi%29\"$ are three consecutive crests (maxima) in the wavy graph, at $\"-3pi%2F2%29%29%29+%2C+%7B%7B%7Bpi%2F2\"$ , and $\"5pi%2F2\"$
\n" ); document.write( "and $\"sin%28-pi%2F2%29=-1=sin%28-pi%2F2%2B2pi%29\"$ are two consecutive minima, at $\"-pi%2F2\"$ and $\"3pi%2F2\"$ .
\n" ); document.write( "In between those extremes, the function takes the same values more often, once on the way up, and again on the way down.
\n" ); document.write( "For example $\"sin%28pi%2F6%29=1%2F2=sin%285pi%2F6%29\"$
\n" ); document.write( " $\"graph%28500%2C100%2C-1%2C9%2C-1.5%2C1.5%2Csin%28x%29%2C1%2F2%29\"$
\n" ); document.write( "With a domain of all the real numbers as possible $\"x\"$ values, the values of $\"f%28x%29=sin%28x%29\"$ repeat, so that function does not have an inverse.
\n" ); document.write( "For that reason, we define $\"arcsin%28x%29\"$ as the inverse function of
\n" ); document.write( " $\"f%28x%29=sin%28x%29\"$ define on the domain $\"-pi%2F2%3C=x%3C=pi%2F2\"$
\n" ); document.write( "In that restricted domain, the only $\"x\"$ that has $\"sin%28x%29=1%2F2\"$ is $\"pi%2F6\"$ .
\n" ); document.write( "The same goes for any other value of $\"sin%28x%29\"$ within the domain defined as $\"-pi%2F2%3C=x%3C=pi%2F2\"$ .
\n" ); document.write( "The other inverse trigonometric functions are defined similarly.
\n" ); document.write( " $\"y=arccos%28x%29\"$ is the angle $\"y\"$ such that
\n" ); document.write( " $\"x=cos%28y%29\"$ and $\"0%3C=y%3C=pi\"$ \n" ); document.write( "

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