document.write( "Question 723880: find all values of feta, to the nearest degree, that satisfy the equation 7cos cubed feta=5cos squared feta+cos feta in the interval 0 degrees to 360 degrees \n" ); document.write( "

Algebra.Com's Answer #443490 by KMST(5328) $\"\"$ $\"About$

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$\"7cos%5E3%28theta%29=5cos%5E2%28theta%29%2Bcos%28theta%29\"$ --> $\"7cos%5E3%28theta%29-5cos%5E2%28theta%29-cos%28theta%29=0\"$
\n" ); document.write( "So that I do not have to keep writing $\"cos%28theta%29\"$ so many times, I will define a shorter-named variable
\n" ); document.write( " $\"x=cos%28theta%29\"$
\n" ); document.write( "Using the newly defined variable I can write the equation as
\n" ); document.write( " $\"7x%5E3-5x%5E2-x=0\"$
\n" ); document.write( "That is much easier to type.
\n" ); document.write( "Now, we solve for x:
\n" ); document.write( " $\"7x%5E3-5x%5E2-x=0\"$ --> $\"x%287x%5E2-5x-1%29=0\"$ (taking out $\"x\"$ as common factor)
\n" ); document.write( "The solutions are $\"x=0\"$ and the solutions to the quadratic equation
\n" ); document.write( " $\"7x%5E2-5x-1=0\"$ , which we can find by using the quadratic formula:
\n" ); document.write( " $\"x+=+%28-%28-5%29+%2B-+sqrt%28%28-5%29%5E2-4%2A7%2A%28-1%29+%29%29%2F%282%2A7%29+\"$
\n" ); document.write( " $\"x+=+%285+%2B-+sqrt%2825%2B28%29%29%2F14\"$
\n" ); document.write( " $\"x+=+%285+%2B-+sqrt%2853%29%29%2F14\"$
\n" ); document.write( "The two values we get from the quadratic formula are approximately
\n" ); document.write( " $\"x=0.877\"$ and $\"x=-0.163\"$ (rounding)
\n" ); document.write( " $\"x=0\"$ corresponds to $\"highlight+%28theta=90%5Eo%29\"$ and $\"highlight+%28theta=270%5Eo%29\"$ for $\"theta\"$ in the interval between $\"0%5Eo\"$ and $\"360%5Eo\"$ , because those are the only two angles with zero cosine in that interval.
\n" ); document.write( "From $\"cos%28theta%29=%285+%2B+sqrt%2853%29%29%2F14\"$ we get $\"highlight+%28theta=28.7%5Eo%29\"$ (rounded) and
\n" ); document.write( " $\"theta=360%5Eo-28.7%5Eo=highlight+%28331.3%5Eo%29\"$ (also a rounded approximation).
\n" ); document.write( "From $\"cos%28theta%29=%285+-+sqrt%2853%29%29%2F14\"$ we get $\"highlight+%28theta=99.4%5Eo%29\"$ (rounded) and
\n" ); document.write( " $\"theta=360%5Eo-99.4%5Eo=highlight+%28260.6%5Eo%29\"$ (also a rounded approximation). \n" ); document.write( "

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