Using the Pythagorean Identity, determine cosθ if sinθ=−12/17 and π<θ<3π/2 .(1 point) Responses 145√17 Start Fraction Start Root 145 End Root over 17 End Fraction 2917−−√ Start Root Start Fraction 29 over 17 End Fraction End Root −145√17 negative Start Fraction Start Root 145 End Root over 17 End Fraction −2917−−√
Using the Pythagorean Identity, we have:
sin^2θ + cos^2θ = 1
Since sinθ = -12/17, we can substitute this value into the equation:
(-12/17)^2 + cos^2θ = 1
144/289 + cos^2θ = 1
cos^2θ = 1 - 144/289
cos^2θ = (289 - 144)/289
cos^2θ = 145/289
Taking the square root of both sides, we get:
cosθ = ± sqrt(145/289)
Since θ is in the third quadrant (π < θ < 3π/2), cosθ is negative. Therefore, the answer is:
cosθ = - sqrt(145/289) = -√(145/17) = -145√17/17