Final answer:
The value of tan when sin = -1/2 and the angle lies in quadrant IV is -√3/3, found by using the Pythagorean identity and noting that cosine is positive in the fourth quadrant.
Step-by-step explanation:
The student asks for the value of tan given that sin = -1/2 and the angle lies in quadrant IV. To find the value of tan when sin = -1/2 in the fourth quadrant, we can use the Pythagorean identity for sine and cosine, knowing that tan(θ) = sin(θ)/cos(θ), and the fact that in the fourth quadrant, sine is negative and cosine is positive.
Firstly, if sin(θ) = -1/2, we can use the Pythagorean identity sin²(θ) + cos²(θ) = 1. Plugging -1/2 into the equation we get (-1/2)² + cos²(θ) = 1, which simplifies to 1/4 + cos²(θ) = 1. Solving for cos²(θ) gives us cos(θ) = √(3)/2, with a positive value in the fourth quadrant.
Therefore, tan(θ) = sin(θ)/cos(θ) = (-1/2)/(√(3)/2), which simplifies to tan(θ) = -1/√3 or tan(θ) = -√3/3.