Final answer:
The cosine of an angle θ with its terminal side in the Third Quadrant will have a negative sign, because both the x and y components in this quadrant are negative.
Step-by-step explanation:
To determine the signs of cos θ when the terminal side of angle θ is in the Third Quadrant, consider that in the context of a right triangle and typical trigonometric definitions, the cosine function relates the length of the adjacent side over the hypotenuse. However, when dealing with angles that place the terminal side in different quadrants of the Cartesian coordinate system, the signs of the trigonometric functions depend on the signs of the x (horizontal) and y (vertical) components of the corresponding points on the unit circle.
In the Third Quadrant, both the x and y components of a point are negative. This means that for an angle θ with its terminal side in the Third Quadrant, cos θ (which corresponds to the x component or adjacent side over hypotenuse) will also be negative. Therefore, in the Third Quadrant, the cosine of any angle will have a negative sign.