Final answer:
To calculate tan theta given that sin(theta) is 1/3 and theta is in quadrant II, we use the Pythagorean identity to find cos(theta) and then divide sin(theta) by cos(theta). Cos(theta) is negative in the second quadrant which leads to tan(theta) being -sqrt(2)/4.
Step-by-step explanation:
To find tan theta with the given information that sin (Theta) = 1/3 and theta being in the second quadrant, we will use the Pythagorean identity for a trigonometric function which states that sin^2(theta) + cos^2(theta) = 1. Since sin(theta) is 1/3, we can solve for cos(theta) by rearranging the identity to cos^2(theta) = 1 - sin^2(theta).
In the second quadrant, the cosine function is negative, so cos(theta) will be the negative square root of the result. Once we find cos(theta), we can calculate tan(theta) which is sin(theta)/cos(theta).
Let's start by finding cos(theta):
- cos^2(theta) = 1 - sin^2(theta) = 1 - (1/3)^2 = 1 - 1/9 = 8/9
- Therefore, cos(theta) = -sqrt(8/9) = -sqrt(8)/3 since cos(theta) should be negative in the second quadrant.
Now we can finally find tan(theta):
- tan(theta) = sin(theta)/cos(theta) = (1/3)/(-sqrt(8)/3) = -1/sqrt(8)
- Which simplifies to tan(theta) = -1/(2*sqrt(2)) = -sqrt(2)/4 after rationalizing the denominator.