Ana was asked to use triangles formed by angles on the unit circle to prove the Pythagorean Identity sin2θ+cos2θ=1 , where sinθ=1617 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
To prove the Pythagorean Identity sin^2θ + cos^2θ = 1, we can use the fact that sinθ = 1617 and π/2 < θ < π.
Since sinθ = opposite/hypotenuse, we can set up a right triangle in the second quadrant where the opposite side is 1617 and the hypotenuse is 1.
Using the Pythagorean Theorem, we can find the adjacent side:
cos^2θ = 1^2 - 1617^2
cos^2θ = 1 - 2607289
cos^2θ = -2607288
Taking the square root of both sides to solve for cosθ:
cosθ = ±√(-2607288)
Since θ is in the second quadrant (π/2 < θ < π), the cosine value will be negative.
cosθ = -√(2607288)
Simplifying the square root gives us:
cosθ = -√(4*65287)
cosθ = -√(4) * √(65287)
cosθ = -2√(65287)
Therefore, Ana should use the value -2√(65287) for cosθ.