Find the cosine for an angle that has a sine of 3√40/20 and is in Quadrant II. Use the Pythagorean identity sin^2(θ)+cos^2(θ)=1 and the quadrant to solve.(1 point) Responses −110 negative Start Fraction 1 over 10 End Fraction −10√10 negative Start Fraction 1 over Start Root 10 End Root End Fraction 110 Start Fraction 1 over 10 End Fraction 10√10
We know that the sine of an angle in Quadrant II is positive.
Since sin(θ) = 3√40/20 = √40/10 = 2√10/10 = √10/5, we can plug in this value into the Pythagorean identity to solve for cos(θ):
sin^2(θ) + cos^2(θ) = 1
(√10/5)^2 + cos^2(θ) = 1
10/25 + cos^2(θ) = 1
cos^2(θ) = 1 - 10/25
cos^2(θ) = 25/25 - 10/25
cos^2(θ) = 15/25
cos^2(θ) = 3/5
Since cos is negative in Quadrant II, we take the negative square root:
cos(θ) = -√(3/5) = -√3/√5 = -√15/5 = -√15/5
Therefore, the cosine of the angle is -√15/5.