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Find i) cos (5 q ) in terms of cos q ii) sin(5 q ) in terms of sin q

Using de Moivres Theorem to find cos n q and sin n q. Find i) cos (5 q ) in terms of cos q ii) sin(5 q ) in terms of sin q iii) tan(5 q ) in terms of tan q. Using De Moivres Theorem. Separating Real and Imaginary Parts. Using Pascals Triangle line 5.

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Find i) cos (5 q ) in terms of cos q ii) sin(5 q ) in terms of sin q

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  1. Using de Moivres Theorem to findcosnq and sin nq Find i) cos(5q) in terms of cosq ii) sin(5q) in terms of sinq iii) tan(5q) in terms of tanq Using De Moivres Theorem Separating Real and Imaginary Parts Using Pascals Triangle line 5 cos(5q) + isin(5q) = (cosq + isinq)5 = cos5q + 5icos4qsinq + 10i2cos3qsin2q+ 10i3cos2qsin3q+ 5i4cosqsin4q+ i5sin5q = cos5q – 10cos3q sin2q + 5cosq sin4q + i(5cos4qsinq – 10cos2qsin3q + sin5q) cos5q= cos5q – 10cos3qsin2q + 5cosqsin4qEquating real parts sin5q= 5cos4qsinq – 10cos2qsin3q + sin5qEquating imaginary parts Replace sin2q by 1– cos2qand sin4q by (1–cos2q)2 in cos5q formula cos5q = cos5q – 10cos3q(1-cos2q) + 5cosq(1-cos2q)2Expand and group terms Replace cos2q by 1 – sin2qand cos4q by (1 – sin2q)2 in sin5q formula sin5q= 5(1–sin2q)2sinq – 10(1–sin2q)sin3q + sin5q Expand and group terms

  2. tan5q = Divide top and bottom terms by cos5q Cancel cos terms

  3. Using de Moivres Theorem to express cosnq and sinnqin terms of cosq and sinq = (cosq+ isinq)-1= cosq - isinq = e–iq z = cosq+isinq = eiq As cos(–q) = cosqand sin(–q) = –sinq zn = cos(nq) + isin(nq) = einq = cosnq – isin(nq) = e–inq

  4. Express cos2q in terms of cosq as = 2cos(2q) + 2 4cos2q = 2cos(2q) + 2 cos2q = (cos(2q) + 1)

  5. 1) Express sin5q in terms of sinq 2) Integrate the answer Pascals Triangle line 5 Grouping powers = 2isin5q – 52isin3q + 102isinq = 2isin5q – 10isin3q + 20isinq

  6. 32isin5q = 2isin5q – 10isin3q + 20isinq sin5q = Hence

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