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Published byIsabella Casson Modified over 3 years ago

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Find i) cos(5 ) in terms of cos ii) sin(5 ) in terms of sin iii) tan(5 ) in terms of tan cos(5 ) + isin(5 ) = (cos + isin ) 5 = cos 5 + 5icos 4 sin + 10i 2 cos 3 sin 2 + 10i 3 cos 2 sin 3 + 5i 4 cos sin 4 + i 5 sin 5 = cos 5 – 10cos 3 sin 2 5cos sin 4 + i(5cos 4 sin – 10cos 2 sin 3 sin 5 cos5 = cos 5 – 10cos 3 sin 2 5cos sin 4 Equating real parts sin5 = 5cos 4 sin – 10cos 2 sin 3 sin 5 Equating imaginary parts Replace sin 2 by 1– cos 2 and sin 4 by (1–cos 2 2 in cos5 formula cos5 = cos 5 – 10cos 3 cos 2 5cos cos 2 Expand and group terms Replace cos 2 by 1 – sin 2 and cos 4 by (1 – sin 2 ) 2 in sin5 formula sin5 = 5(1–sin 2 sin – 10(1–sin 2 sin 3 sin 5 Expand and group terms Using de Moivres Theorem to find cos n and sin n Using De Moivres Theorem Using Pascals Triangle line 5 Separating Real and Imaginary Parts

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tan5 = Divide top and bottom terms by cos 5 Cancel cos terms

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Using de Moivres Theorem to express cos n and sin n in terms of cos and sin z = cos + isin e i z n = cos(n + isin(n e in = (cos + isin = cos isin e –i = cosn – isin(n e –in As cos(– = cos and sin(– ) = –sin

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Express cos 2 in terms of cos = 2cos(2 ) + 2 as 4cos 2 = 2cos(2 ) + 2 cos 2 = (cos(2 ) + 1)

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1) Express sin 5 in terms of sin 2) Integrate the answer = 2isin5 – 5 2isin3 + 10 2isin = 2isin5 – 10isin3 + 20isin Pascals Triangle line 5 Grouping powers

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32isin 5 = 2isin5 – 10isin3 + 20isin sin 5 = Hence

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