Final answer:
The correct value for tan θ when cos θ = 1/4 and sin θ < 0 is -√15, which is not reflected in any of the provided answer options. The available options may be incorrect or incomplete.
Step-by-step explanation:
To find tan θ when cos θ = 1/4 and sin θ < 0, we need to understand the relation between trigonometric functions. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, which can also be expressed using sine and cosine as tan θ = sin θ / cos θ.
Since we already have cos θ = 1/4, we need to find sin θ. We can use the Pythagorean identity, which states that sin2 θ + cos2 θ = 1. Knowing that cos θ = 1/4, we can find sin θ as follows:
sin2 θ = 1 - cos2 θ = 1 - (1/4)2
Calculating this gives us:
sin2 θ = 1 - 1/16 = 15/16
sin θ can be either positive or negative, but since we know that sin θ < 0, sin θ must be the negative square root:
sin θ = -√(15/16)
Finally, we calculate tangent:
tan θ = sin θ / cos θ = -√(15/16) / (1/4) = -4 √(15/16)
As we are looking for the value of tan θ among the given options, which are integers or simple fractions, it seems we must simplify:
tan θ = -4 √(15)/4 = -√15
The given options do not reflect this simplified result. However, the negative sign tells us that the answer will be negative. Looking at the options, the only negative value is option 3) -1/4, which does not match the correct calculation. So, based on the calculations, none of the provided options are correct. The student may need to review the question or options provided.