2sin^2 A + 5cosA = 4 , Prove that cos A = 1/2
2sin^2 A + 5cosA = 4
2(1 - cos^2 A) + 5cosA - 4 = 0
2cos^2 A - 5cosA + 2 = 0
(2cosA -1)(cosA - 2) = 0
cosA = 1/2 or cosA = -2, the latter is not possible,
so cosA = 1/2
To prove that cos A = 1/2, we can rewrite the given equation using the trigonometric identity: sin^2 A + cos^2 A = 1.
First, let's substitute cos^2 A with (1 - sin^2 A):
2sin^2 A + 5cos A = 4
2sin^2 A + 5(1 - sin^2 A) = 4
Simplifying:
2sin^2 A + 5 - 5sin^2 A = 4
Now combine like terms:
-3sin^2 A + 5 = 4
Rearranging the terms:
-3sin^2 A = -1
Dividing both sides by -3:
sin^2 A = 1/3
Taking the square root of both sides:
sin A = ±sqrt(1/3)
Since -1 ≤ sin A ≤ 1, we discard the negative value and keep the positive value.
sin A = sqrt(1/3)
Now, we can use the relationship between sine and cosine to find cos A:
cos^2 A = 1 - sin^2 A
cos^2 A = 1 - (1/3)
cos^2 A = 2/3
Taking the square root of both sides:
cos A = ±sqrt(2/3)
Again, we discard the negative value since -1 ≤ cos A ≤ 1 and keep the positive value.
cos A = sqrt(2/3)
Therefore, cos A is not equal to 1/2 as required.