Final answer:
sin(θ₁) in the third quadrant, given that cos(θ₁) = -13/15, can be determined using the Pythagorean identity. The sine value is found to be negative in this quadrant and calculating it results in -4/15 * √(7/3).
Step-by-step explanation:
The student is asking about an angle located in the third quadrant and the cosine value of that angle is given as -13/15. To find the sine value of an angle when the cosine value is known, we can use the Pythagorean identity which states that sin²(θ) + cos²(θ) = 1 for any angle θ.
Since the angle is in the third quadrant, we know that sine values are also negative here. Given cos(θ₁) = -13/15, first, we find sin²(θ₁) using the identity: sin²(θ₁) = 1 - cos²(θ₁) = 1 - (-13/15)². We then take the square root to find sin(θ₁), remembering to assign the negative sign because we are in the third quadrant.
Calculating this gives us sin(θ₁) = -√(1 - (-13/15)²) = -√(1 - 169/225) = -√(56/225) = -√(8*7/15*15) = -4/15 * √(7/3). Thus, the value of sin(θ₁) is -4/15 * √(7/3).