Prove that cos40cos80cos160 is equal to -1/8.
Answers
Step-by-step explanation:
Let`s use an angle wich we know the cosine, i. e. 120º
Let be the angle a=40º
The formula for the triple angle is:
cos(3a) = 4(cos(a))^3 - 3cos(a)
We know cos(120)=-1/2.
cos(120) = 4(cos(40))^3 - 3cos(40) = -1/2
Then for x=cos(a), we have a third grade polynomial in x:
4x^3 - 3x + 1/2 = 0
One of the roots of this polynomial will be equal to cos(40)
Factorizing we obtain:
(x + 0.94)(x - 0.766)(x - 0.174) = 0
The roots are -0.94, 0.766 and 0.174
cos(40) is positive and greater than cos(45), then:
cos(40) = 0.766
The formula for double angle is
cos(2a) = 2(cos(a))^2 - 1, then
cos(80) = 2(0.766)^2 -1
cos(80) = 0.174
Using the same formula:
cos(160) = 2(0.174)^2 - 1
cos(160) = -0.94
Then cos(40)*cos(80)*cos(160) = 0.766 * 0.174 * -0.94
= -0.125=-1/8