Final answer:
Given that sinθ = 2/3, the possible values for cosine and tangent are cosθ = -√5/3 and tanθ = 2/√5, as these adhere to the trigonometric functions' relationships in a right triangle. Option 2 is correct.
Step-by-step explanation:
When given that sinθ = 2/3, we can find the corresponding cosine and tangent values by using Pythagorean identities and the definitions of trigonometric functions in a right triangle.
Firstly, using the Pythagorean identity sin^2θ + cos^2θ = 1, we can solve for cosθ:
sin^2θ = (2/3)^2 = 4/9
cos^2θ = 1 - sin^2θ = 1 - 4/9 = 5/9
cosθ = ±√(5/9) = ±√5/3
Since cosθ can be either positive or negative, depending on the quadrant, we look at the possible answers:
cosθ = -√5/3 is possible if θ is in the second or third quadrant.
cosθ = √5/3 is only possible if θ is in the first quadrant, which is not the case here as secθ cannot be negative when cosθ is positive.
Now, let's determine tanθ using tanθ = sinθ / cosθ:
For cosθ = -√5/3, tanθ = (2/3) / (-√5/3) = -2/√5
As a result, the pair cosθ = -√5/3 and tanθ = 2/√5 is the correct answer since the signs of cosine and tangent must agree in the second and third quadrants.