document.write( "Question 515399: If cosα=4/5 and cosβ=12/13, find sin(α+β) when α and β are the measures of two first-quadrant angles. Give answer as a fraction. \n" ); document.write( "

Algebra.Com's Answer #343950 by drcole(72) $\"\"$ $\"About$

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To find $\"sin%28alpha+%2B+beta%29\"$ , we need to use the sum formula for sine:\r
\n" ); document.write( "\n" ); document.write( " $\"sin%28alpha+%2B+beta%29+=+sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29\"$ \r
\n" ); document.write( "\n" ); document.write( "We have $\"cos%28alpha%29+=+4%2F5\"$ and $\"cos%28beta%29+=+12%2F13\"$ , and we know that $\"alpha\"$ and $\"beta\"$ are first quadrant angles, which implies that both $\"sin%28alpha%29\"$ and $\"sin%28beta%29\"$ are positive. To find $\"sin%28alpha%29\"$ , recall the trigonometric identity:\r
\n" ); document.write( "\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+%2B+%28cos%28alpha%29%29%5E2+=+1\"$ \r
\n" ); document.write( "\n" ); document.write( "Using what we know, we can solve for $\"sin%28alpha%29\"$ :\r
\n" ); document.write( "\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+%2B+%284%2F5%29%5E2+=+1\"$ (substituting $\"4%2F5\"$ for $\"cos%28alpha%29\"$ )
\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+%2B+16%2F25+=+1\"$ (simplifying)
\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+=+1+-+16%2F25\"$ (subtracting $\"16%2F25\"$ from both sides)
\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+=+25%2F25+-+16%2F25\"$ (putting all terms on the right side under a common denominator)
\n" ); document.write( " $\"%28sin%28alpha%29%29%5E2+=+9%2F25\"$ (simplifying)
\n" ); document.write( " $\"sin%28alpha%29+=+3%2F5\"$ (taking the positive square root of both sides, since we know $\"sin%28alpha%29\"$ is positive)\r
\n" ); document.write( "\n" ); document.write( "Using the same sort of calculation, we get that $\"sin%28beta%29+=+5%2F13\"$ . So now we substitute into the sum formula:\r
\n" ); document.write( "\n" ); document.write( " $\"sin%28alpha+%2B+beta%29+=+sin%28alpha%29%2Acos%28beta%29+%2B+cos%28alpha%29%2Asin%28beta%29\"$
\n" ); document.write( " $\"sin%28alpha+%2B+beta%29+=+%283%2F5%29%2A%2812%2F13%29+%2B+%284%2F5%29%2A%285%2F13%29\"$ (substituting)
\n" ); document.write( " $\"sin%28alpha+%2B+beta%29+=+36%2F65+%2B+20%2F65\"$ (simplifying)
\n" ); document.write( " $\"sin%28alpha+%2B+beta%29+=+56%2F65\"$ (simplifying)
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