Unformatted text preview: + j sin ± kπ N ¶¶ , expressing your answer in Cartesian form. (Use the identity 1+2+ ··· + n = n ( n +1) / 2 to simplify the answer ﬁrst.) S 2.2 . Show that the expression f ( θ ) = ej 2 θejθ + 1e jθ + e j 2 θ , where θ ranges over [0 , 2 π ), is realvalued, and obtain an alternative expression for it in terms of sines and/or cosines. S 2.3 . Clearly, e j 3 θ = ‡ e jθ · 3 By expanding (cos θ + j sin θ ) 3 and separating real and imaginary parts, obtain two identities: one for cos3 θ in terms of powers of cos θ only, and another for sin3 θ in terms of powers of sin θ only. S 2.4 (more diﬃcult). Consider two complex numbers w and z , where w 6 = 0 is ﬁxed and z is variable such that  z  = 1. Show that as z traces out the unit circle, the ratio  zw *   z(1 /w )  is constant in value. ( Hint : If  z  = 1, then 1 /z = z * .)...
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 Spring '08
 staff
 Trigonometry, pts, Complex number, Euler's formula, Complex Plane

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