document.write( "Question 696991: Find the real root of the transcedental equation cos x - 3x + 1 = 0 correct to four decimal places using iteration method. \n" ); document.write( "

Algebra.Com's Answer #429650 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
NOTE: In math, when we see $\"cos%28x%29\"$ , we assume $\"x\"$ is measured in radians, not degrees.
\n" ); document.write( "
\n" ); document.write( " $\"cos+x+-+3x+%2B+1+=+0\"$ <--> $\"cos+x+%2B+1+=+3x\"$
\n" ); document.write( "We know that the graphs of $\"y=cos%28x%29%2B1\"$ and $\"y=3x\"$ look like this:
\n" ); document.write( " $\"graph%28300%2C150%2C-2%2C7%2C-2%2C2.5%2Ccos%28x%29%2B1%2C3x%29\"$
\n" ); document.write( "We need to look for the point where they intersect.
\n" ); document.write( "
\n" ); document.write( "At $\"x=0.5\"$ , $\"y=cos%28x%29%2B1\"$ is greater than $\"y=3x\"$ ,
\n" ); document.write( " $\"cos%280.5%29%2B1=about\"$ $\"1.8776%3E1.5=3%2A0.5\"$ ,
\n" ); document.write( "but at $\"x=0.7\"$ , $\"y=cos%28x%29%2B1\"$ is lesser than $\"y=3x\"$ ,
\n" ); document.write( " $\"cos%280.7%29%2B1=about\"$ $\"1.7648%3C2.1=3%2A0.7\"$ ,
\n" ); document.write( "so the answer is between $\"x=0.5\"$ and $\"x=0.7\"$ .
\n" ); document.write( "For $\"x=0.6\"$ , $\"y=cos%28x%29%2B1\"$ is greater than $\"y=3x\"$ , with
\n" ); document.write( " $\"cos%280.6%29%2B1=about\"$ $\"1.8253%3E1.8=3%2A0.6\"$ ,
\n" ); document.write( "but they are very close, much closer that at $\"x=0.7\"$ ,
\n" ); document.write( "so the answer is between $\"x=0.6\"$ and $\"x=0.7\"$ ,
\n" ); document.write( "but we expect it to be close to $\"x=0.6\"$ .
\n" ); document.write( "For $\"x=0.61\"$ , $\"y=cos%28x%29%2B1\"$ is lesser than $\"y=3x\"$ , with
\n" ); document.write( " $\"cos%280.61%29%2B1=about\"$ $\"1.8196%3C1.83=3%2A0.61\"$ ,
\n" ); document.write( "so the answer is between $\"x=0.6\"$ and $\"x=0.61\"$ .
\n" ); document.write( "We may try $\"x=0.605%7D%7D%2C+getting%0D%0A%7B%7B%7Bcos%280.605%29%2B1=about\"$ $\"1.8225%3E1.815=3%2A0.605\"$ ,
\n" ); document.write( "where $\"y=cos%28x%29%2B1\"$ is still a bit too high,
\n" ); document.write( "and then try $\"x=0.607\"$ , getting
\n" ); document.write( " $\"cos%280.607%29%2B1=about\"$ $\"1.8214%3E1.8210=3%2A0.607\"$ ,
\n" ); document.write( "where $\"y=cos%28x%29%2B1\"$ is still a bit too high,
\n" ); document.write( "but very, very close.
\n" ); document.write( "We could next try $\"x=0.6071\"$ , getting
\n" ); document.write( " $\"cos%280.6071%29%2B1=about\"$ $\"1.82131%3E1.8213=3%2A0.6071\"$ ,
\n" ); document.write( "where $\"y=cos%28x%29%2B1\"$ is still a hair too high.
\n" ); document.write( "We can then try $\"x=0.6072\"$ , getting
\n" ); document.write( " $\"cos%280.6072%29%2B1=about\"$ $\"1.8212%3E1.8216=3%2A0.6072\"$ ,
\n" ); document.write( "where $\"y=cos%28x%29%2B1\"$ is a bit too low.
\n" ); document.write( "That tells us that the answer is between $\"x=0.6071\"$ and $\"x=0.6072\"$ .
\n" ); document.write( "We seemed to be closer at $\"x=0.6071\"$ , so maybe we should try $\"x=0.60714\"$ .
\n" ); document.write( "For $\"x=0.60714\"$ , $\"y=cos%28x%29%2B1\"$ is also a bit too low, with
\n" ); document.write( " $\"cos%280.60714%29%2B1=about\"$

and $\"x=0.60714\"$ ,
\n" ); document.write( "both of which round to $\"highlight%280.6071%29\"$ . \n" ); document.write( "

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