document.write( "Question 825018: find the exact values of sin(A+B), cos(A+B), and tan(A+B), given:\r
\n" ); document.write( "\n" ); document.write( "a.) sin A = 8/17, tan B = 5/12, A and B in Q1
\n" ); document.write( "b.) cos A = -12/13, cot B= 24/7, A in Q2, B in Q3
\n" ); document.write( "c.) sin A = 1/3, sin B = 2/5, A in Q1, B in Q2 \n" ); document.write( "

Algebra.Com's Answer #496987 by Edwin McCravy(20059) $\"\"$ $\"About$

You can put this solution on YOUR website!
find the exact values of sin(A+B), cos(A+B), and tan(A+B), given:\r
\n" ); document.write( "\n" ); document.write( "a.) sin A = 8/17, tan B = 5/12, A and B in Q1
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\r\n" );
document.write( "Draw the picture of angle A in the first quadrant Q1, and since \r\n" );
document.write( "the sine is opposite/hypotenuse or y/r, and since we are given \r\n" );
document.write( "sin A = 8/17, make y=8 and r=17, and calculate x using the \r\n" );
document.write( "Pythagorean theorem:\r\n" );
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document.write( "x²+y² = r²\r\n" );
document.write( "x²+8² = 17²\r\n" );
document.write( "x²+64 = 289\r\n" );
document.write( "   x² = 225\r\n" );
document.write( "    x = √225\r\n" );
document.write( "    x = 15\r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Draw the picture of angle B in the first quadrant Q1, and since \r\n" );
document.write( "the tangent is opposite/adjacent or y/x, and since we are given \r\n" );
document.write( "tan B = 5/12, make y=5 and x=12, and calculate r using the \r\n" );
document.write( "Pythagorean theorem:\r\n" );
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document.write( " x²+y² = r²\r\n" );
document.write( "12²+5² = r²\r\n" );
document.write( "144+25 = r²\r\n" );
document.write( "   169 = r²\r\n" );
document.write( "  √169 = r\r\n" );
document.write( "    13 = r\r\n" );
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document.write( "\r\n" );
document.write( "Now we can solve the problem using the above graphs, \r\n" );
document.write( "with sine = y/r, cosine = x/r and tangent = y/x, and\r\n" );
document.write( "the identities:\r\n" );
document.write( "\r\n" );
document.write( "sin(A+B) = sin(A)cos(B)+cos(A)sin(B)\r\n" );
document.write( "sin(A+B) = \r\n" );
document.write( "sin(A+B) =  \r\n" );
document.write( "sin(A+B) = \r\n" );
document.write( "\r\n" );
document.write( "cos(A+B) = cos(A)cos(B)-sin(A)sin(B)\r\n" );
document.write( "cos(A+B) = \r\n" );
document.write( "cos(A+B) =  \r\n" );
document.write( "cos(A+B) = \r\n" );
document.write( "\r\n" );
document.write( "tan(A+B) =  =  =  =  =  \r\n" );
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document.write( "---------------------------------------------------\r\n" );
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document.write( "b.) cos A = -12/13, cot B= 24/7, A in Q2, B in Q3\r\n" );
document.write( "\r\n" );
document.write( "Draw the picture of angle A in the second quadrant Q2, and since \r\n" );
document.write( "the cosine is adjacent/hypotenuse or x/r, and since we are given \r\n" );
document.write( "cos A = -12/13, make x=-12 and r=13, and calculate y using the \r\n" );
document.write( "Pythagorean theorem:\r\n" );
document.write( "\r\n" );
document.write( "    x²+y² = r²\r\n" );
document.write( "(-12)²+y² = 13²\r\n" );
document.write( "   144+y² = 169\r\n" );
document.write( "       y² = 25\r\n" );
document.write( "        y = √25 = r\r\n" );
document.write( "        y = 5 \r\n" );
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document.write( "We take y positive because it goes up from the x-axis, \r\n" );
document.write( "and we take x negative because it goes to the left of \r\n" );
document.write( "the origin on the x-axis:\r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Draw the picture of angle B in the third quadrant Q3, and since \r\n" );
document.write( "the cotangent is adjacent/opposite or x/y, and since we are given \r\n" );
document.write( "cot B = 24/7, make x=-24 and y=-7, and calculate r using the \r\n" );
document.write( "Pythagorean theorem.  We take y negative because it goes down,\r\n" );
document.write( "and we take x negative because it goes left.\r\n" );
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document.write( "       x²+y² = r²\r\n" );
document.write( "(-7)²+(-24)² = r²\r\n" );
document.write( "      49+576 = r²\r\n" );
document.write( "         625 = r²\r\n" );
document.write( "        √625 = r\r\n" );
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document.write( "(r is always taken positive)         \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we can solve the problem using the above graphs, \r\n" );
document.write( "with sine = y/r, cosine = x/r and tangent = y/x, and\r\n" );
document.write( "the identities:\r\n" );
document.write( "\r\n" );
document.write( "sin(A+B) = sin(A)cos(B)+cos(A)sin(B)\r\n" );
document.write( "sin(A+B) = \r\n" );
document.write( "sin(A+B) =  \r\n" );
document.write( "sin(A+B) = \r\n" );
document.write( "\r\n" );
document.write( "cos(A+B) = cos(A)cos(B)-sin(A)sin(B)\r\n" );
document.write( "cos(A+B) = \r\n" );
document.write( "cos(A+B) = \r\n" );
document.write( "cos(A+B) =  \r\n" );
document.write( "cos(A+B) = \r\n" );
document.write( "\r\n" );
document.write( "tan(A+B) =  =  =  =  =  \r\n" );
document.write( "\r\n" );
document.write( "---------------------------------------------------\r\n" );
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document.write( "c.) sin A = 1/3, sin B = 2/5, A in Q1, B in Q2\r\n" );
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document.write( "You can do that one yourself.  Draw the angles in \r\n" );
document.write( "the correct quadrants. Use the Pythagorean \r\n" );
document.write( "theorem to get the third sides.  You will do it the \r\n" );
document.write( "same way as the other two except the x's will have \r\n" );
document.write( "to be left as square roots instead of whole numbers, \r\n" );
document.write( "as the first two came out to be. For angle A, x will\r\n" );
document.write( "be √8 = √4·2 = 2√2.  For angle B, x will be -√21. \r\n" );
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document.write( "Edwin

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