Final answer:
To calculate sin2(theta) given that tan(theta)=8/9 and cos(theta) > 0, we use the Pythagorean identity and the double angle formula for sine. The answer is sin2(theta) = 64/145.
Step-by-step explanation:
To find sin^2(theta), we'll work with the given information that tan(theta) = 8/9 and that cos(theta) > 0.
Since tan(theta) is positive and cos(theta) is positive, theta must be in the first quadrant.
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1 and the fact that tan(theta) = sin(theta)/cos(theta),
we can deduce the values of sin(theta) and cos(theta).
First, if tan(theta) = 8/9, we can imagine a right triangle where the opposite side (sin(theta)) is 8 and the adjacent side (cos(theta)) is 9.
The hypotenuse (r) can be found using the Pythagorean theorem: r^2 = 8^2 + 9^2 => r = √(64 + 81) = √145.
Therefore, sin(theta) = 8/√145 and cos(theta) = 9/√145.
Finally, using the identity for double angle sine, which is sin(2*theta) = 2*sin(theta)*cos(theta), we get:
sin(2*theta) = 2 * (8/√145) * (9/√145) = 144/145.
So, the value of sin^2(theta) is (8/√145)^2 = 64/145.