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 −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
To find the cosine of an angle when given the sine and quadrant, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, along with the information that the angle is in quadrant II.
Given that sin(θ) = 340√20, we can divide both sides by 340 to get:
sin(θ)/340 = √20
Squaring both sides, we have:
(sin(θ)/340)^2 = 20
Simplifying, we get:
sin^2(θ)/340^2 = 20
Multiplying both sides by 340^2, we have:
sin^2(θ) = 20 * 340^2
Taking the square root of both sides, we get:
sin(θ) = ± √(20 * 340^2)
Since the angle is in quadrant II, sin(θ) is positive. Hence, we have:
sin(θ) = √(20 * 340^2)
Now, we can use the Pythagorean identity to find the cosine. Rearranging the identity, we have:
cos^2(θ) = 1 - sin^2(θ)
Substituting in the value of sin(θ), we get:
cos^2(θ) = 1 - (√(20 * 340^2))^2
Simplifying, we have:
cos^2(θ) = 1 - 20 * 340^2
Taking the square root of both sides, we get:
cos(θ) = ± √(1 - 20 * 340^2)
Since the angle is in quadrant II, cos(θ) is negative. Hence, we have:
cos(θ) = -√(1 - 20 * 340^2)
Therefore, the cosine of the angle is -√(1 - 20 * 340^2).