Ana was asked to use triangles formed by angles on the unit circle to prove the Pythagorean Identity sin^2 θ+cos^2 θ=1 , where sinθ=16/17 and π2<θ<π . What value should she use for cosθ ?(1 point) Responses 33√17 Start Fraction Start Root 33 End Root over 17 End Fraction 117√ Start Fraction 1 over Start Root 17 End Root End Fraction −117√ negative Start Fraction 1 over Start Root 17 End Root End Fraction −33√17
The Pythagorean Identity states that sin^2 θ + cos^2 θ = 1.
Given that sinθ = 16/17 and π/2 < θ < π, we can determine the value of cosθ using the Pythagorean Identity.
To determine cosθ, we can use the fact that sinθ = opposite/hypotenuse and cosθ = adjacent/hypotenuse. In this case, the opposite side is 16 and the hypotenuse is 17.
Using the Pythagorean Theorem in this right triangle, we can find the adjacent side:
adjacent^2 + 16^2 = 17^2
adjacent^2 + 256 = 289
adjacent^2 = 289 - 256
adjacent^2 = 33
Taking the square root of both sides, we find that adjacent = √33.
Therefore, cosθ = adjacent/hypotenuse = √33/17.
So, the value that Ana should use for cosθ is √33/17. Answer: 33√17/17.