Final answer:
To find the exact trigonometric value when cotθ=-√5/2, we use the Pythagorean Identity and the relationship between tangent, sine, and cosine to solve for sinθ and cosθ. Since the angle is in the second quadrant, the sine value will be positive, and the cosine will be negative.
Step-by-step explanation:
To determine the exact trigonometric value when given that cotθ=-√5/2, and π/2<θ, we first note that the cotangent function is the reciprocal of the tangent function. Therefore, if cotθ=-√5/2, it implies that tanθ=-2/√5. Since π/2<θ, we're dealing with an angle in the second quadrant where the sine function is positive and the cosine function is negative, leading to a negative tangent value.
To find the exact values of the sine and cosine functions, we can use the Pythagorean Identity, sin^2θ + cos^2θ = 1. Knowing tanθ, we can express sinθ and cosθ in terms of tangent and secant (reciprocal of cosine) and solve for the exact values.
Since tanθ = sinθ/cosθ, we can write sinθ as tanθ*cosθ. To find cosθ, we use the identity 1+tan^2θ = sec^2θ, which allows us to solve for cosθ and subsequently sinθ, yielding the exact trigonometric values.