Subtracting and Multiplying Sines of Angle MultiplesDate: 04/01/2010 at 10:30:17 From: amos Subject: trigonometric Using 18 degrees as x, prove that sin 54 - sin 18 = 1/2 and sin 54 * sin 18 = 1/4 I have tried proving it but I can't go beyond (cos2x - sinx)/2 Because 18 is exactly one-third of 54, I know that sin 54 - sin 18 = 2cos 2x sin x. cos 2x = sin 3x 2sin 3x sin x = -(cos 4x - cos 2x) cos 4x = sin x sin 3x sin x = (cos 2x - sin x)/2 Help me out. Thank you. Date: 04/01/2010 at 13:02:55 From: Doctor Robert Subject: Re: trigonometric Hello, What is the original problem? I don't see why the cos is brought into play when you are asked to solve sin54 - sinx = 1/4 sin54 - 1/4 = sinx sin x = .5590 x = 33.99 degrees - Doctor Robert, The Math Forum http://mathforum.org/dr.math/ Date: 04/13/2010 at 12:31:05 From: amos Subject: trigonometric The problem is not to solve, it is to PROVE that sin 54 - sin 18 = 1/2 Thank you very much. Date: 04/15/2010 at 10:00:47 From: Doctor Jacques Subject: Re: trigonometric Hi Amos, You want to prove that: sin 54 - sin 18 = 1/2 sin 54 * sin 18 = 1/4 First, let us show that the two propositions are equivalent. Use the formula: sin p - sin q = 2*sin ((p - q)/2) cos ((p + q)/2) with p = 54. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 04/16/2010 at 04:25:12 From: amos Subject: trigonometric Please can you go further solving this question? sin 72 = 2sin 36 sin 54, I keep mixing it up. Thank you. Date: 04/16/2010 at 06:23:21 From: Doctor Jacques Subject: Re: trigonometric Hi Amos, Once you find that, you can substitute sin 72 into equation [2] sin 36 = 2 sin 18 sin 72 [2] This substitution gives sin 36 = 2 sin 18 (2 sin 36 sin 54) = 4 sin 18 sin 36 sin 54 [4] And, if you divide both sides of [4] by (4 sin 36), you obtain: 1/4 = sin 18 sin 54 This is the second equation, which we wanted to prove. As explained before, the first equation is a consequence of it. Does this make sense? - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 04/27/2010 at 08:49:30 From: amos Subject: Thank you (trigonometric) Thank you so much. I am so grateful. You really helped me. God bless you. |
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