Question

tính cos 10 độ.cos30 độ.cos50 độ.cos70 độ

Answer 1

Answer:

3/ 16

Step-by-step explanation:

cos(10°)cos(30°)cos(50°)cos(70°) 

= [cos(70°)cos(10°)][cos(50°)cos(30°)] 

= (1/2)(1/2)[cos(80°) + cos(60°)][cos(80°) + cos(20°)], from above 

= (1/4)[cos^2(80°) + cos(80°)cos(20°) + cos(80°)cos(60°) + cos(60°)cos(20°)]. 

Using the above formula again: 

(1/4)[cos^2(80°) + cos(80°)cos(20°) + cos(80°)cos(60°) + cos(60°)cos(20°)] 

= (1/4)(1/2)[cos(160°) + cos(100°) + cos(60°) + cos(140°) + cos(20°) + cos(80°) + cos(40°) + 1] 

= (1/8)[cos(160°) + cos(100°) + cos(60°) + cos(140°) + cos(20°) + cos(80°) + cos(40°) + 1] 

= (1/8)[cos(160°) + cos(100°) + cos(140°) + cos(20°) + cos(80°) + cos(40°)] + 3/16. 

We now want to show that: 

cos(160°) + cos(100°) + cos(140°) + cos(20°) + cos(80°) + cos(40°) = 0. 

To do this, re-arrange the terms as follows: 

[cos(160°) + cos(20°)] + [cos(140°) + cos(40°)] + [cos(100°) + cos(80°)]. 

Using the sum-to-product formula: 

cos(A)cos(B) = 2cos[(A + B)/2]cos[(A - B)/2], 

each bracketed term equals zero (as (A + B)/2 = 90° and cos(90°) = 0), so this equals: 

0 + 0 + 0 = 0, as required. 

Therefore, cos(10°)cos(30°)cos(50°)cos(70°) = (1/8)(0) + 3/16 = 3/16.