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 Question from citra, a student: sin54 cos36/cos18 - 2cos36 -2sin18 = ... thank you very much..

Hi Citra.

Did you notice that all of those numbers are multiples of 18?

So if I can show you how to find sin 18°, you should be able to use that in conjunction with double angle identities and sum identities to find the whole thing.

[ credit for the following: http://www.andrews.edu/~calkins/math/webtexts/NUMB18.HTM#SIN18 ]

 sin 72° = 2 sin 36° cos 36° by the double angle relationship. sin 72° = 4 sin 18° cos 18° (1 - 2sin2 18°) by the double angle relationship, again. cos 18° = 4 sin 18° cos 18° (1 - 2sin2 18°) by the cofunction properties: sin 72° = cos 18°. 1 = 4 sin 18° (1 - 2sin2 18°) Let x = sin 18°, this is known as 1 = 4x(1-2x2) substitution, a useful technique in calculus. 8x3-4x+1 = 0 A product is zero only when one of its factors is zero. 8x3-4x+1 = (2x-1)(4x2+2x-1)=0 (2x-1)=0 implies x= ½=sin 30° > sin 18° ; Since we know sin is increasing on [0°,90°]. x = (-2 ± √(4 + 4•4•1))/8 So we must solve the other factor, = (-2 ± √20)/8 using the quadratic formula. = (-2 ± √4 √5)/8 = (-1 ± √5)/4 But the sin 18° > 0, so it cannot be negative. sin 18° = (√ 5 - 1) / 4 Hence the middle root is the one we want.

If you consider the unit circle, this means sin 18° is shown on a triangle of radius 1, with the vertical side going up to (√ 5 - 1) / 4. Pythagoras can tell you what the horizontal side is, which is cos 18°.

Now try using double angle formulas and addition (3*18 = 2*18 + 18) formulas to expand:

sin(3*18°) cos(2*18°) / cos(18°) - 2 cos(2*18°) - 2 sin(18°).

Then you'll have the exact value of the expression you sent to us.

Cheers,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.