Bài 28 trang 205 SGK Giải tích 12 Nâng cao, Viết các số phức sau dưới dạng lượng giác...

Bài 28. Viết các số phức sau dưới dạng lượng giác:

(eqalign{
& a),,1 – isqrt 3 ;,,1 + i;,,(1 – isqrt 3 )(1 + i);,,{{1 – isqrt 3 } over {1 + i}}; cr
& b),,2ileft( {sqrt 3 – i} ight); cr
& c),,{1 over {2 + 2i}}; cr
& d),,z = sin varphi + icos varphi ,(varphi inmathbb R) cr} )

Giải

(eqalign{
& a),,1 – isqrt 3 = 2left( {{1 over 2} – {{sqrt 3 } over 2}i} ight) = 2left( {cos left( { – {pi over 3}} ight) + isin left( { – {pi over 3}} ight)} ight);,,,,, cr
& ,,,,,,,,1 + i = sqrt 2 left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i} ight) = sqrt 2 left( {cos left( {{pi over 4}} ight) + isin left( {{pi over 4}} ight)} ight);, cr
& ,,,,,,,,(1 – isqrt 3 )(1 + i) = 2sqrt 2 left( {{1 over 2} – {{sqrt 3 } over 2}i} ight)left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i} ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left( {cos left( { – {pi over 3}} ight) + isin left( { – {pi over 3}} ight)} ight)left( {cos {pi over 4} + isin {pi over 4}} ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left[ {cos left( {{pi over 4} – {pi over 3}} ight) + isin left( {{pi over 4} – {pi over 3}} ight)} ight] cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left[ {cos left( { – {pi over {12}}} ight) + isin left( { – {pi over {12}}} ight)} ight];,, cr
& {{1 – isqrt 3 } over {1 + i}} = sqrt 2 left[ {cos left( { – {pi over 3} – {pi over 4}} ight) + isin left( { – {pi over 3} – {pi over 4}} ight)} ight] cr
& ,,,,,,,,,,,,,,,,,,,;;;, = sqrt 2 left[ {cos left( { – {7 over {12}}pi } ight) + isin left( { – {7 over {12}}pi } ight)} ight]; cr
& b),,2i = 2left( {cos {pi over 2} + isin {pi over 2}} ight) cr
& ,,,,,,,left( {sqrt 3 – i} ight) = 2left( {{{sqrt 3 } over 2} – {1 over 2}i} ight) = 2left[ {cos left( { – {pi over 6}} ight) + isin left( { – {pi over 6}} ight)} ight]; cr
& ,,,,,,,2ileft( {sqrt 3 – i} ight) = 4left[ {cos left( {{pi over 2} – {pi over 6}} ight) + isin left( {{pi over 2} – {pi over 6}} ight)} ight] cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ;;,= 4left[ {cos left( {{pi over 3}} ight) + isin left( {{pi over 3}} ight)} ight] cr
& c),,2 + 2i = 2sqrt 2 left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i} ight) = 2sqrt 2 left( {cos {pi over 4} + isin {pi over 4}} ight), cr
& Rightarrow {1 over {2 + 2i}} = {1 over {2sqrt 2 }}left[ {cos left( { – {pi over 4}} ight) + isin left( { – {pi over 4}} ight)} ight] cr  
& d),z = ,sin varphi + icos varphi = ,cos left( {{pi over 2} – varphi } ight) + isinleft( {{pi over 2} – varphi } ight)(varphi in mathbb R) cr} )