Find the exact value of sin 2teta=2 sin teta cos teta given that cos teta =2/3, 0<teta< pi/2
To find the exact value of sin 2theta given that cos theta = 2/3, we can use the double angle identity for sine.
The double angle identity for sine states that sin 2theta = 2 sin theta cos theta.
Given that cos theta = 2/3, we can plug this value into the double angle identity:
sin 2theta = 2 sin theta (2/3).
To find the exact value of sin theta, we can use the Pythagorean identity sin^2 theta + cos^2 theta = 1.
Since cos theta = 2/3, we can square it: (2/3)^2 = 4/9.
And then subtract it from 1 to get sin^2 theta: 1 - 4/9 = 5/9.
Taking the square root of both sides, we find that sin theta = √(5/9) = √5/3.
Now, we can substitute this value back into the initial equation:
sin 2theta = 2 (√5/3) (2/3).
Simplifying further, we have:
sin 2theta = (4√5/9).
Therefore, the exact value of sin 2theta, given that cos theta = 2/3, is (4√5/9).