When tan θ = 4 and θ is in the third quadrant, cos θ is negative. Using the Pythagorean identity, we find that sec θ = -√17, which is approximately -4. Thus, the final answer is sec θ = -4, corresponding to option B.
To find sec θ when tan θ = 4 and θ lies in the third quadrant, we use the identity sec θ = 1/cos θ. Recall that in the third quadrant, both sine and cosine are negative, but tangent is positive since it is the ratio of sine to cosine. Therefore, cos θ must be negative in this context since tan θ is positive.
We can determine cos θ using the Pythagorean identity: 1 + tan² θ = sec² θ. Substituting tan θ = 4, we get sec² θ = 1 + 4² = 17. Thus, sec θ = ±√17. Since we are in the third quadrant where sec θ (or cos θ) is negative, we choose sec θ = -√17.
The answer choices are in decimal or fractional form, so we should also convert √17 to a decimal which is approximately 4.1231. Hence, the closest match in the answer choices is -4, which we find by inverting cos θ and assigning the appropriate sign based on the quadrant.
Final answer: The value of sec θ is -4 (Option B).