Question
Find the cosine for an angle that has a sine of 340√20 and is in Quadrant II. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 and the quadrant to solve.(1 point)
Responses
−10√10
negative Start Fraction 1 over Start Root 10 End Root End Fraction
110
Start Fraction 1 over 10 End Fraction
10√10
Start Fraction 1 over Start Root 10 End Root End Fraction
−110
Responses
−10√10
negative Start Fraction 1 over Start Root 10 End Root End Fraction
110
Start Fraction 1 over 10 End Fraction
10√10
Start Fraction 1 over Start Root 10 End Root End Fraction
−110
Answers
GPT 3.5
The sine of an angle in Quadrant II is always positive. Since the sine of the angle is given as 340√20, which is positive, we can conclude that the sine of the angle is positive.
Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can solve for cos(θ):
(340√20)^2 + cos^2(θ) = 1
115600(20) + cos^2(θ) = 1
2312000 + cos^2(θ) = 1
cos^2(θ) = 1 - 2312000
cos^2(θ) = -2311999
Since the cosine of an angle can't be negative, we know that there is no real solution for cos(θ). Therefore, none of the given options are correct.
Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can solve for cos(θ):
(340√20)^2 + cos^2(θ) = 1
115600(20) + cos^2(θ) = 1
2312000 + cos^2(θ) = 1
cos^2(θ) = 1 - 2312000
cos^2(θ) = -2311999
Since the cosine of an angle can't be negative, we know that there is no real solution for cos(θ). Therefore, none of the given options are correct.