document.write( "Question 174442: 36. sin(u-v) sin u=5/13
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Algebra.Com's Answer #129508 by solver91311(24713)\"\" \"About 
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It is a little difficult to decipher what you are asking here. I think you mean find the exact value of \"sin%28u-v%29\" given that \"sin%28u%29=5%2F13\" and \"cos%28v%29=-3%2F5\"\r
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\n" ); document.write( "\n" ); document.write( "First of all, you need the difference identity: \"sin%28alpha-beta%29=sin%28alpha%29cos%28beta%29-cos%28alpha%29sin%28beta%29\".\r
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\n" ); document.write( "\n" ); document.write( "Your problem is that you now need to determine \"cos%28u%29\" given that \"sin%28u%29=5%2F13\" and \"sin%28v%29\" given that \"cos%28v%29=-3%2F5\".\r
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\n" ); document.write( "\n" ); document.write( "Knowing that \"sin%28u%29%3E0\", you are certain that \"u\" is an angle in the First or Second Quadrants. Remembering that a 5-12-13 triangle is a right triangle and the fact that the sine function is defined as opposite/hypotenuse, we can see directly that \"cos%28u%29=12%2F13\" (if \"u\" is First Quadrant) or \"cos%28u%29+=+-12%2F13\" (if \"u\" is Second Quadrant).\r
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\n" ); document.write( "\n" ); document.write( "Similarly, \"v\" must be Quadrant II or III because \"cos%28v%29%3C0\" and 3-4-5 is a right triangle, so \"sin%28v%29=4%2F5\" or \"sin%28v%29=-4%2F5\"\r
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\n" ); document.write( "\n" ); document.write( "All you need to do is plug the values into \"sin%28alpha-beta%29=sin%28alpha%29cos%28beta%29-cos%28alpha%29sin%28beta%29\" or, in the case of this problem \"sin%28u-v%29=sin%28u%29cos%28v%29-cos%28u%29sin%28v%29\", and do the arithmetic. Because there are two possible values for each of the derived single angle functions, you will have 4 different calculations to perform, however you will end up with two pairs of identical results -- hence you will have two answers in the end. \n" ); document.write( "
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