document.write( "Question 541147: Find an acute angle θ that satisfies the equation.
\n" ); document.write( "sin⁡θ=cos⁡2θ+60°)
\n" ); document.write( "tip: (sin⁡θ=cos⁡(90-θ))
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Algebra.Com's Answer #354107 by KMST(5328) $\"\"$ $\"About$

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$\"sin%28theta%29=+cos%282theta%2B60%29\"$
\n" ); document.write( "With the tip, it transforms into
\n" ); document.write( " $\"cos%2890-theta%29=cos%282theta%2B60%29\"$
\n" ); document.write( "If two angles are the same, they have the same cosine.
\n" ); document.write( "So at least we'll get a solution from $\"90-theta=2theta%2B60\"$
\n" ); document.write( "Adding $\"theta\"$ to both sides, we get $\"90=3theta%2B60\"$
\n" ); document.write( "Subtracting 60 from both sides, we get $\"30=3theta\"$
\n" ); document.write( "Dividing both sides by 3, we get $\"10=theta\"$
\n" ); document.write( "So $\"theta=10degrees\"$ is a solution.
\n" ); document.write( "Are there others?
\n" ); document.write( "Could the cosines be equal, but the angles be different?
\n" ); document.write( "In general, that could happen, but since $\"theta\"$ is an acute angle
\n" ); document.write( " $\"0%3Ctheta%3C90\"$ --> $\"90-0%3E90-theta%3E90-90\"$ --> $\"90%3E90-theta%3E0\"$
\n" ); document.write( "So 90- $\"theta\"$ ; is an acute angle too. That means its cosine is a positive number.
\n" ); document.write( " $\"0%3Ctheta%3C90\"$ --> $\"0%3C2theta%3C180\"$ ---> $\"60%3C2theta%2B60%3C240\"$
\n" ); document.write( "The angle $\"2theta%2B60\"$ seems to have more options, but since its cosine is equal to a positive number, it is more restricted.
\n" ); document.write( "The only cosines that area positive for angles between 60° and 240° are those for angles between 60° and 90°. Between 90° and 270° they are negative.
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