第十章 复数 章末知识梳理（学案）-【成才之路】2021-2022学年高中新教材数学必修第四册新课程同步学习指导（人教B版）

新教材·高中新课程学习指导 　 　 知识点 3　 r1 r2 [cos(θ1 - θ2) + isin(θ1 - θ2)]　 除以　 减去 关键能力·攻重难 题型探究 　 　 典例 1:(1)2 cos 2π3 + isin 2π 3( )× 3 cos 5π 6 + isin 5π 6( ) = 2 3 cos 3π2 + isin 3π 2( )= - 2 3i. (2)2(cos 5° + isin 5°) ×4(cos 30° + isin 30°) × 12 (cos 25° + isin 25°) = 8(cos 35° + isin 35°) × 12 (cos 25° + isin 25°) = 4(cos 60° + isin 60°) = 2 + 2 3i. 　 　 对点练习 1:(1)2i　 法一:( 3 + i)(cos 60° + isin 60°) = 2(cos 30° + isin 30°)(cos 60° + isin 60°) = 2(cos 90° + isin 90°) = 2i. 法二:( 3 + i)(cos 60° + isin 60°) = ( 3 + i) 1 2 + 3 2 i( ) = 32 + 3 2 i + 1 2 i - 3 2 = 2i. (2) z2 = 2(cos 150° - isin 150°) = 2[cos( - 150°) + isin( - 150°)], ∴ z1 z2 = 8 × 2[cos(240° - 150°) + isin(240° - 150°)] = 16(cos 90° + isin90°) = 16i. 　 　 典例 2:8 cos 7π6 + isin 7π 6( )÷ 4 cos π 3 + isin π 3( )[ ] = 2 cos 5π6 + isin 5π 6( )= 2 - 3 2 + 1 2 i( )= - 3 + i. 　 　 对点练习 2:(1)2i ÷ 12 (cos 30° + isin 30°)[ ] = 2(cos 90° + isin 90°) ÷ 12 (cos 30° + isin 30°)[ ] = 4(cos 60° + isin 60°) = 2 + 2 3i. (2)i3 ÷ 12 (cos 120° + isin 120°)[ ]= - i ÷ 1 2 (cos 120° + isin 120°)[ ] = (cos 270° + isin 270°) ÷ 12 (cos 120° + isin 120°)[ ] = 2[cos(270° - 120°) + isin(270° - 120°)] = 2(cos 150° + isin 150°) = - 3 + i. 　 　 典例 3:欲求∠Z2OZ1,可计算 z1 z2 . ∵ z1 z2 =1 +2 3i 7 + 3i = (1 +2 3i)(7 - 3i) (7 + 3i)(7 - 3i) =1 + 3i4 = 1 2 cos π 3 + isin π 3( ), ∴ ∠Z2OZ1 = π 3 ,且 |OZ1 → | |OZ2 → | = 1 2 , 由余弦定理,设 | OZ1 | = k, | OZ2 | = 2k( k > 0),则 | Z1Z2 | 2 = k2 + (2k) 2 - 2k·2k·cos π3 = 3k 2,∴ |Z1Z2 | = 3k,而 k2 + ( 3k) 2 = (2k) 2, ∴ △OZ1Z2 为有一角为 60°的直角三角形. 　 　 对点练习 3:依题意知( - 1 + 3i)· cos 4π3 + isin 4π 3( ) = z2 cos 3π4 + isin 3π 4 . ∴ z2 = ( - 1 + 3i) cos 4π 3 + isin 4π 3( ) cos 3π 4 + isin 3π 4( ) = 2· cos 2π3 + 4π 3 + 3π 4( )+ isin 2π 3 + 4π 3 + 3π 4( )[ ] = 2 cos 11π4 + isin 11π 4( )= - 2 + 2i. 易错警示 　 　 典例 4: z = 1 + cos θ + isin θ = 1 + 2cos 2 θ2 - 1( ) + 2i·sin θ 2 · cos θ2 = 2cos θ 2 cos θ 2 + isin θ 2( ). ∵ π < θ < 2π,∴ π2 < θ 2 < π,∴ cos θ 2 < 0, ∴ 2cos θ2 cos θ 2 + isin θ 2( )= - 2cos θ 2 - cos θ 2 - isin θ 2( ) = - 2cos θ2 cos π +