Final answer:
The cosine value provided is incorrect as it exceeds the maximum cosine function value. However, with a valid cosine value and the condition that sinθ < 0, the sine value can be found using the Pythagorean identity. The negative square root is taken due to sinθ being less than zero.
Step-by-step explanation:
The given condition is cosθ = 3√3 with the additional information that sinθ < 0. However, the cosine value seems incorrect as the maximum value for the cosine function is 1, therefore cosθ = 3√3 cannot be true. Assuming there is a typo and considering the correct range for cosine, the answer can be derived using the Pythagorean identity:
cos2θ + sin2θ = 1.
Since sinθ < 0, it indicates that the angle θ is in either the third or fourth quadrant. In both quadrants, cosine values can still be positive. After getting the correct cosine value within the range of -1 to 1, you would find sinθ by rearranging the Pythagorean identity:
sin2θ = 1 - cos2θ.
Then, take the square root and apply the negative sign since sinθ < 0.