Final answer:
To find the exact value of tan θ given that sin θ = -8/9 and tan θ > 0, we determine that θ is in the third quadrant where both sine and cosine are negative. Using the Pythagorean identity, we find cos θ = -√17/9, and then tan θ = 8√17/17 by dividing the sine by the cosine.
Step-by-step explanation:
The student's question is asking to find the exact value of a trigonometric function given certain conditions about sine and tangent. Specifically, it is given that sin θ is -8/9 and tan θ is positive. Since the sine is negative and the tangent is positive, θ must be in the third quadrant where both sine and cosine are negative (since tan θ = sin θ/cos θ, and a negative divided by a negative is a positive).
From the known value of the sine function, we can find the cosine function using the Pythagorean identity:
∑cos θ = ±√(1 - sin^2 θ) = ±√(1 - (-8/9)^2) = ±√(1 - 64/81) = ±√(17/81) = ±√17/9. Since we are in the third quadrant, cosine is also negative, so cos θ = -√17/9.
Now we can find tan θ using the definition tan θ = sin θ/cos θ:
tan θ = (-8/9) / (-√17/9) = 8/√17. To rationalize the denominator, we multiply numerator and denominator by √17, giving us tan θ = 8√17/17.