cosA=1/2 than cos4A=?
cos 4A
= cos^2 2A - sin^2 2A
= (cos 2A - sin 2A)(cos 2A + sin 2A)
= (cos^2 A - sin^2 A - 2sinAcosA)(cos^2 A - sin^2 A + 2sinAcosA)
from cosA = 1/2 ----> sinA = √3/2
(A = 60° or π/3)
plug the cos and sin values into my last equation and evaluate, I will leave the arithmetic up to you
cosA = 1/2
A = π/3
4A = 4π/3
cos4A = -1/2
Nice!!!
Well, I always like to take those long detours.
I once went from Toronto to Niagara Falls by going all the way around Lake Ontario.
As Crocodile Dundee would say,
Now that's a detour! (eh?)
To find the value of cos 4A, we can use the multiple angle formula for cosine. The multiple angle formula for cosine states that:
cos(2θ) = cos²θ - sin²θ
Using this formula, we can find the value of cos 4A by repeatedly applying the double angle formula:
cos 4A = cos(2*(2A))
= cos²(2A) - sin²(2A)
To further simplify, we can express cos²(2A) and sin²(2A) in terms of cos(2A) and sin(2A):
cos²(2A) = (1 + cos(4A))/2
sin²(2A) = (1 - cos(4A))/2
Substituting these expressions into the equation:
cos 4A = (1 + cos(4A))/2 - (1 - cos(4A))/2
= 1/2 + cos(4A)/2 - 1/2 + cos(4A)/2
= cos(4A)
Therefore, we can conclude that cos 4A is equal to cos(4A).