using double angle formulas,find the values of the following expressions in surd form.
1) sin105°
2)sin(3π/8)
3) By considering cos 36°=(√5+1)/4, express sin18 in surd form.
the model ans is (√5-1)/4 but I can only find √{[1-(√5+1)/4]/2} which is the same value.
三題數關於double angle
2011-05-10 6:16 am
回答 (2)
2011-05-10 7:44 am
✔ 最佳答案
1 sin105= sin(90 + 15)
= cos15
= √[(cos30+1)/2]
= √[(√3 + 2)/4]
2 sin(3π/8)
= cos(π/2 - 3π/8)
= cos(π/8)
= √[cos(π/4) + 1)/2]
= √[(√2 + 2)/4]
3 sin18
= √[(1 - cos36)/2]
= √[(4 - √5 - 1)/8]
= √[(3 - √5)/8]
= √[(√5 - 1)^2/16]
= (√5 - 1)/4
2011-05-10 7:41 am
1)sin105°
=sin(60+45)°
= sin60cos45+sin45cos60
=(√3)/2*(√2)/2+(√2)/2*1/2
=(√6+√2)/4
2)sin(3π/8)
=√((1-cos(3π/4))/2)
=√(1/2+(cos(π/4))/2)
=√(1/2+(√2)/4)
=√((√2+2)/4)
3) cos 36°=(√5+1)/4
1-2(sin18)^2=(√5+1)/4
sin18=√((3-√5)/8)
sin18=√(6-2√5)/4
sin18=√(1-2√5+5)/4
sin18=√((√5-1)^2)/4
sin18= (√5-1)/4
=sin(60+45)°
= sin60cos45+sin45cos60
=(√3)/2*(√2)/2+(√2)/2*1/2
=(√6+√2)/4
2)sin(3π/8)
=√((1-cos(3π/4))/2)
=√(1/2+(cos(π/4))/2)
=√(1/2+(√2)/4)
=√((√2+2)/4)
3) cos 36°=(√5+1)/4
1-2(sin18)^2=(√5+1)/4
sin18=√((3-√5)/8)
sin18=√(6-2√5)/4
sin18=√(1-2√5+5)/4
sin18=√((√5-1)^2)/4
sin18= (√5-1)/4
收錄日期: 2021-04-26 14:55:23
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20110509000051KK01213