document.write( "Question 1056110: Find all the angles between 0° and 360° inclusive which satisfy the equation cot(135°-2y)=-0.5 \n" ); document.write( "
Algebra.Com's Answer #671317 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"cot%28%22135%B0%22-2y%29\"\"%22%22=%22%22\"\"-0.5\"\r\n" );
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document.write( "Cotangent is the reciprocal of the tangent:\r\n" );
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document.write( "\"1%2Ftan%28%22135%B0%22-2y%29\"\"%22%22=%22%22\"\"-0.5\"\r\n" );
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document.write( "Multiply both sides by tan(135°-2y):\r\n" );
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document.write( "\"1\"\"%22%22=%22%22\"\"-0.5tan%28%22135%B0%22-2y%29\"\r\n" );
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document.write( "Divide both sides by -0.5:\r\n" );
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document.write( "\"1%2F%28-0.5%29\"\"%22%22=%22%22\"\"tan%28%22135%B0%22-2y%29\"\r\n" );
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document.write( "Simplify left side:\r\n" );
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document.write( "\"-2\"\"%22%22=%22%22\"\"tan%28%22135%B0%22-2y%29\"\r\n" );
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document.write( "Use formula for tan(A-B):\r\n" );
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document.write( "\"-2\"\"%22%22=%22%22\"\"%28tan%28%22135%B0%22%29-tan%282y%29%29%2F%281%2Btan%28%22135%B0%22%29tan%282y%29%29\"\r\n" );
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document.write( "Use the fact that tan(135°) = -1, substituting:\r\n" );
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document.write( "\"-2\"\"%22%22=%22%22\"\"%28%28-1%29-tan%282y%29%29%2F%281%2B%28-1%29tan%282y%29%29\"\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "\"-2\"\"%22%22=%22%22\"\"%28-1-tan%282y%29%29%2F%281-tan%282y%29%29\"\r\n" );
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document.write( "Multiply both sides by 1-tan(2y):\r\n" );
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document.write( "\"-2%281-tan%282y%29%5E%22%22%29\"\"%22%22=%22%22\"\"-1-tan%282y%29\"\r\n" );
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document.write( "Distribute on the left:\r\n" );
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document.write( "\"-2%2B2tan%282y%29\"\"%22%22=%22%22\"\"-1-tan%282y%29\"\r\n" );
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document.write( "Add +2 to both sides:\r\n" );
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document.write( "\"2tan%282y%29\"\"%22%22=%22%22\"\"1-tan%282y%29\"\r\n" );
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document.write( "Add + tan(2y) to both sides:\r\n" );
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document.write( "\"3tan%282y%29\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Divide both sides by 3:\r\n" );
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document.write( "\"tan%282y%29\"\"%22%22=%22%22\"\"1%2F3\"\r\n" );
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document.write( "Use formula for tan(2θ):\r\n" );
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document.write( "\"2tan%5E%22%22%28y%29%2F%281-tan%5E2%28y%29%29\"\"%22%22=%22%22\"\"1%2F3\"\r\n" );
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document.write( "Cross-multiply:\r\n" );
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document.write( "\"6tan%5E%22%22%28y%29\"\"%22%22=%22%22\"\"1-tan%5E2%28y%29\"\r\n" );
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document.write( "Get 0 on the right:\r\n" );
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document.write( "\"tan%5E2%28y%29%2B6tan%5E%22%22%28y%29-1\"\"%22%22=%22%22\"\"0\"\r\n" );
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document.write( "Use quadratic formula:\r\n" );
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document.write( "\"tan%28y%29+=+%28-6+%2B-+sqrt%28+6%5E2-4%2A1%2A%28-1%29+%29%29%2F%282%2A1%29+\"\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "\"tan%28y%29+=+%28-6+%2B-+sqrt%2836%2B4%29%29%2F2+\"\r\n" );
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document.write( "\"tan%28y%29+=+%28-6+%2B-+sqrt%2840%29%29%2F2+\"\r\n" );
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document.write( "\"tan%28y%29+=+%28-6+%2B-+sqrt%284%2A10%29%29%2F2+\"\r\n" );
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document.write( "\"tan%28y%29+=+%28-6+%2B-+2sqrt%2810%29%29%2F2+\"\r\n" );
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document.write( "\"tan%28y%29+=+%282%28-3+%2B-+sqrt%2810%29%29%29%2F2+\"\r\n" );
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document.write( "\"tan%28y%29+=+%28cross%282%29%28-3+%2B-+sqrt%2810%29%29%29%2Fcross%282%29+\"\r\n" );
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document.write( "\"tan%28y%29+=+-3+%2B-+sqrt%2810%29+\"\r\n" );
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document.write( "Using the + :\r\n" );
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document.write( "\"tan%28y%29+=+-3+%2B+sqrt%2810%29+\"\r\n" );
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document.write( "\"tan%28y%29+=+0.1622776602\"\r\n" );
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document.write( "\"y+=+%229.217474413%B0%22\"  <-- 1st quadrant solution\r\n" );
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document.write( "Add 180° to get the 3rd quadrant solution:\r\n" );
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document.write( "\"y+=+%22189.217474413%B0%22\"  <-- 3rd quadrant solution\r\n" );
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document.write( "Using the - :\r\n" );
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document.write( "\"tan%28y%29+=+-3+-+sqrt%2810%29+\"\r\n" );
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document.write( "\"tan%28y%29+=+-6.1622776602\"\r\n" );
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document.write( "\"y+=+%22-80.78252559%B0%22\"  <-- 4th quadrant negative solution\r\n" );
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document.write( "which calculator always gives but which we cannot accept here\r\n" );
document.write( "since it's not between 0° and 360°.\r\n" );
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document.write( "So we add 360° to get an equivalent positive 4th quadrant \r\n" );
document.write( "solution which is between 0° and 360:\r\n" );
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document.write( "\"y+=+%22279.217474413%B0%22\"  <-- 4th quadrant solution\r\n" );
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document.write( "Subtract 180° to get the 2nd quadrant solution:\r\n" );
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document.write( "\"y+=+%2299.217474413%B0%22\"  <-- 2nd quadrant solution\r\n" );
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document.write( "So there are 4 solutions between 0° and 360°.\r\n" );
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document.write( "Edwin
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