anonymous
  • anonymous
Evaluate the expression using an appropriate double angle formula. cos^2(15) - sin^2(15)
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
cos^2(15) - sin^2(15) = cos 2*15 = cos 30 = (sqrt3)/2
anonymous
  • anonymous
use of the identtiy cos 2x = cos^2x - sin^2x
anonymous
  • anonymous
How? could you please explain in detail please?

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anonymous
  • anonymous
there is an established formula: cos 2x = cos^2x - sin^2x which is true for all angles x the reverse is also true: cos^2x - sin^2x = cos2x so if you substitute x = 15 you get cos^2(15) - sin^2(15) = cos(2*15) cos(2*15) = cos 30 now you can use a calculator to find the cosine of 30 you get 0.866 which is a valid answer - i used a special right angled triangle
anonymous
  • anonymous
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anonymous
  • anonymous
oh got it ! thanks ! and how about this one, sin(105)cos(105) ? do i use sin2x = 2sinxcosx ? but how do i use that one ?
anonymous
  • anonymous
yes sin 2x = 2 sinx cosx so sinx cos x =sin 2x/ 2 =( sin 210) / 2 to find sin 210 look up sin 210-180 = 30 on your calculator this gives 0.5 now 210 degrees is in third quadrant where sine is negative so sin 210 = -0.5 so sin(105)cos(105) = sin 210 /2 = -0.5 / 2 = -0.25
anonymous
  • anonymous
could you check if this is correct, i just did it. 2sin(pi/8)cos(pi/8) 2sinx cosx = sin2x 2sin(pi/8) cos(pi/8)= sin2(pi/8) sin2(pi/8) = sin (1/4 pi) = 0.0137 is it correct ?
anonymous
  • anonymous
sinx cos x =sin 2x/ 2 =( sin 210) / 2 in this part, why did you divide it by 2 ?
anonymous
  • anonymous
2sinscosx = sin2x divide both sides by 2 sinx cos x = sin2x / 2 your original expression was sin(105)cos(105)
anonymous
  • anonymous
thanks a lot ! and the question i just did by myself, is it correct ?
anonymous
  • anonymous
correct up to last line sin pi/4 - 0.707 did you set the calculator to radians? - it might have been in degrees
anonymous
  • anonymous
yeah - i just checked - you must have set it ti degrees to get .0137
anonymous
  • anonymous
oh i didnt change it to radians. i just changed it to radians, my answer is 0.7071
anonymous
  • anonymous
right
anonymous
  • anonymous
could you help me out with this question ? 6tan(75) / 1-tan^2 (75)
anonymous
  • anonymous
sorry - i would if i had time - gotta go out
anonymous
  • anonymous
oh alright, thanks a lot for your help !

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