document.write( "Question 762223: Suppose that cos(t)=5/13 where t is in the fourth quadrant. Find sin(2t) \n" ); document.write( "
Algebra.Com's Answer #519684 by Theo(13342)\"\" \"About 
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if cos(t) = 5/13, then t = arccos(5/13) = 67.38013505 if t is in Q1.
\n" ); document.write( "t will be equal to 360 - 67.38013505 if t is in Q4.
\n" ); document.write( "that means that t = 292.6198649 degrees.
\n" ); document.write( "there are several ways to find cos(2t).
\n" ); document.write( "the first way is to double the angle and then find its cosine.
\n" ); document.write( "when t = 292.6198649 degrees, 2t = 585.2397299 degrees.
\n" ); document.write( "cos(585.2397299) = -.7041420118
\n" ); document.write( "the second way is to use the formula that tells you that cos(2t) = cos^2t - sin^2t.
\n" ); document.write( "this is derived from the formula for cos(a + b) = cos(a)cos(b) - sin(a)sin(b).
\n" ); document.write( "if a is equal to t and b is equal to t, then this formula becomes cos(t)cos(t) - sin(t)sin(t) which becomes cos^2(t) - sin^2(t).
\n" ); document.write( "since sin^2(t) is equal to 1 - cos^2(t), then this formula becomes cos^2(t) - (1 - cos^2(t) which becomes cos^2(t) - 1 + cos^2(t) which becomes 2cos^2(t) - 1.
\n" ); document.write( "the formula is therefore cos(2t) = 2cos^2(t) - 1.
\n" ); document.write( "applying this formula to t, we get:
\n" ); document.write( "cos(2t) = 2cos^2(t) - 1 becomes cos(2*292.6198649) = 2cos^2(292.6198649) - 1 which becomes 2*(cos(292.6198649))^2 - 1 which becomes 2*(.3846153846)^2 - 1 which becomes -.7041420118.
\n" ); document.write( "your answer should be that cos(2t) = -.7041420118.
\n" ); document.write( "there are other ways to derive it but these are the ones that i am familiar with so far.
\n" ); document.write( "i'm still experimenting with the first way to determine if it will always find the correct answer.
\n" ); document.write( "so far it's been doing good, but the verdict is not in yet, at least not for me.
\n" ); document.write( "if in doubt, go with the second way.
\n" ); document.write( "that is the method using trigonometric identities that is normally taught.\r
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