document.write( "Question 607411: If cos(x)=-1/4 and tan(X) is negative what is Sin2x \n" ); document.write( "
Algebra.Com's Answer #382682 by jsmallt9(3759)\"\" \"About 
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I'm assuming that an exact expression is desired. The first solution will have that exact solution. The second solution may be simpler but it will be a decimal approximation of the answer.

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  • Since sin(2x) = 2*sin(x)*cos(x) and since we already know cos(x), all we need to find is sin(x) in order to be able to find sin(2x).
    • To find sin(x) we can use \"sin%5E2%28x%29+=+1+-+cos%5E2%28x%29\":
    \n" ); document.write( "\"sin%5E2%28x%29+=+1+-+%28-1%2F4%29%5E2\"
    \n" ); document.write( "\"sin%5E2%28x%29+=+1+-+1%2F16\"
    \n" ); document.write( "\"sin%5E2%28x%29+=+15%2F16\"
    \n" ); document.write( "\"sqrt%28sin%5E2%28x%29%29+=+sqrt%2815%2F16%29\"
    \n" ); document.write( "\"sin%28x%29+=+0+%2B-+sqrt%2815%29%2Fsqrt%2816%29\"
    \n" ); document.write( "\"sin%28x%29+=+0+%2B-+sqrt%2815%29%2F4\"
    \n" ); document.write( "(NOTE: The zeroes are not necessary. The reason they are there is that algebra.com's software will not allow a \"plus or minus\" symbol without having something in front of it. So please ignore the zeroes.)
    \n" ); document.write( "Now we just have to figure out whether to use the positive or negative value. To decide this we need to know which quadrant in which x terminates...
  • From the fact that both cos(x) and tan(x) are both negative we can determine the quadrant in which x terminates. cos is negative in the 2nd and 3rd quadrants while tan is negative in the 2nd and 4th quadrants. So for both cos and tan to be negative the angle must terminate in the second quadrant.
  • Since x terminates in the 2nd quadrant and since sin is positive there, we will be using the positive value for sin(x): \"sqrt%2815%29%2F4\"
  • We are now ready to find sin(2x):
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    \n" ); document.write( "which is the exact value for sin(2x).

\n" ); document.write( "For a decimal approximation we can...
  • Again determine that x terminates in the 2nd quadrant.
  • Use \"cos%5E%28-1%29%281%2F4%29+=+75.52248781\" to determine that the reference angle is 75.52248781 degrees.
  • An angle in the 2nd quadrant with a reference of 75.52248781 degrees is 180 - 75.52248781 = 104.47751219 degrees. So x = 104.47751219 degrees.
  • This makes 2x = 2*104.47751219 = 208.95502437 degrees.
  • From our calculator we can find the sin(2x) = sin(208.95502437) = -0.48412292

\n" ); document.write( "NOTE: If you use your calculator on \"-sqrt%2815%29%2F8\" you will find that our second answer is indeed close to the exact solution of \"-sqrt%2815%29%2F8\". \n" ); document.write( "
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