trigonometric function m2

用戶31980

2011-06-19 5:04 am

prove that (cos10degree)(cos30degree)(cos50degree)(cos70degree)=3/16

更新1:

in fact, what is your logic to think of this sequence?:) can you teach me

回答 (1)

myisland8132

2011-06-19 5:35 am

✔ 最佳答案

cos10°cos30°cos50°cos70°

= cos70°cos10°cos50°cos30°

= (1/2)[cos(70°+10°) + cos(70°-10°)]cos50°(√3/2)

= (√3/4)[cos80° + cos60°]cos50°

【Using identity cosa cosb = (1/2)[cos(a+b) + cos(a-b)]】

= (√3/4)[cos80°cos50° + cos60°cos50°]

= (√3/4)【(1/2)[cos(80°+50°) + cos(80°-50°)] + (1/2)cos50°】

= (√3/4)【(1/2)[cos130° + cos30°] + (1/2)cos50°】

= (√3/4)【(1/2)[cos(180°-50°) + cos30°] + (1/2)cos50°】

【Using identity cosa cosb = (1/2)[cos(a+b) + cos(a-b)]】

= (√3/4)【(1/2)[-cos50° + √3/2] + (1/2)cos50°】

= (-√3/8)cos50° + 3/16 + (√3/8)cos50°

= 3/16

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