Final answer:
Given cscθ = -√26/5 and cosθ < 0, sinθ is -5√26/26 after rationalization, and cotθ is √26/5, due to the negative cosine and sine values leading to a positive cotangent in the second quadrant.
Step-by-step explanation:
To find sinθ and cotθ, we start with the given that cscθ = -√26/5. Since the cosecant is the reciprocal of the sine function, we have sinθ = -5/√26. We can rationalize the denominator to get sinθ = -5√26/26.
For cotθ, which is the reciprocal of tanθ, we need to use the information cosθ < 0. We know that sinθ is also negative because cscθ is negative. Therefore, tanθ, being the ratio of sinθ to cosθ, is positive because both are negative (in the second quadrant). Since cotθ is the reciprocal of tanθ, cotθ must be positive: cotθ = cosθ/sinθ.
Given the Pythagorean identity sin^2θ + cos^2θ = 1, we can solve for cosθ. First, we find sin^2θ = 25/26, thus cos^2θ = 1 - 25/26 = 1/26, and since cosθ is negative, cosθ = -√1/√26 or cosθ = -√26/26. Now, cotθ = cosθ/sinθ = (-√26/26) * (-√26/5) = √26/5.
Hence, the correct answer is sinθ = -5√26/26 and cotθ = √26/5.