Answer:To find the exact value of sin(theta), we can use the Pythagorean identity sin^2(theta) + cos^2(theta) = 1. We can also use the fact that tan(theta) = sin(theta) / cos(theta), and that theta is in quadrant 2, where sin(theta) is positive and cos(theta) is negative. Here are some steps you can follow:
Given that tan(theta) = -2/9, we can write sin(theta) = -2k and cos(theta) = -9k, where k is a positive constant. This way, we can express sin(theta) and cos(theta) in terms of tan(theta).
To find the value of k, we can plug in sin(theta) and cos(theta) into the Pythagorean identity and solve for k. We get (-2k)^2 + (-9k)^2 = 1, which simplifies to 85k^2 = 1, and then k = 1 / √85.
To find the exact value of sin(theta), we can substitute k = 1 / √85 into sin(theta) = -2k. We get sin(theta) = -2 / √85.
To simplify the answer, we can rationalize the denominator by multiplying both the numerator and the denominator by √85. We get sin(theta) = -2√85 / 85.
Therefore, the exact value of sin(theta) is Option 1: -2√85 / 85.
Explanation: