If sinA = 1/3 and cosA<0
find the exact value of tan2A
sinA = 1/3 and cosA<0
find the exact value of tan2A
What quadrant is A? Sin is +, cos is - so quadrant 2, so 2 A is quadrant 3 or 4
cos^2A = 1 - sin^2A = 1-1/9 = 8/9
cos A = -(2/3)sqrt 2
tanA = (1/3)/[-(2/3)sqrt2)] = -1/(2 sqrt 2)
tan 2A = 2 [-1/(2 sqrt 2)]/[1- 1/8]
To find the exact value of tan2A, we'll use the double-angle identity for tangent:
tan2A = (2tanA) / (1 - tan^2A)
Given that sinA = 1/3 and cosA < 0, we can use the Pythagorean identity to find the value of tanA. Here's how:
Since sinA = 1/3, we know that the opposite side of angle A is 1 and the hypotenuse is 3. Using the Pythagorean theorem, we can find the adjacent side:
cosA = sqrt(1 - sin^2A)
cosA = sqrt(1 - (1/3)^2)
cosA = sqrt(1 - 1/9)
cosA = sqrt(8/9)
cosA = -sqrt(8)/3 (since cosA is negative)
So, in a right-angled triangle with angle A, we have:
opposite side = 1
adjacent side = -sqrt(8)/3
hypotenuse = 3
Now, we can find the value of tanA by dividing the opposite side by the adjacent side:
tanA = opposite / adjacent
tanA = (1) / (-sqrt(8)/3)
tanA = -3sqrt(8)/8
Now, we can substitute this value of tanA into the formula for tan2A:
tan2A = (2tanA) / (1 - tan^2A)
tan2A = (2 * (-3sqrt(8)/8)) / (1 - (-3sqrt(8)/8)^2)
Simplifying further:
tan2A = -3sqrt(8)/4 / (1 - 9/64)
To divide by a fraction, we multiply by its reciprocal:
tan2A = -3sqrt(8)/4 * (64/55)
tan2A = -12sqrt(8)/11
So, the exact value of tan2A is -12sqrt(8)/11.