Find the exact value of costheta if ,sintheta=-2/3 and theta is in standard position with its terminal side in Quadrant IV.
cos θ = ± √ ( 1 - sin² θ )
In quadrant IV sine is negative , cosine is positive.
So in quadrant IV cos θ = √ ( 1 - sin² θ )
In this case:
cos θ = √ ( 1 - sin² θ ) = √ [ 1 - ( - 2 / 3 )² ] = √ ( 1 - 4 / 9 ) =
√ ( 9 / 9 - 4 / 9 ) = √ ( 5 / 9 ) = √ 5 / √ 9 = √ 5 / 3
in QIV,
y = -2
r = 3
x = √5
cosθ = x/r = √5/3
To find the exact value of cos(theta), we have the value of sin(theta) and the information that theta is in standard position with its terminal side in Quadrant IV.
In Quadrant IV, both x and y coordinates are positive. Since sin(theta) = -2/3, we can use the Pythagorean identity to find the value of cos(theta):
cos(theta) = √(1 - sin^2(theta))
Substituting sin(theta) = -2/3:
cos(theta) = √(1 - (-2/3)^2)
= √(1 - 4/9)
= √(5/9)
= √5/√9
= √5/3
Therefore, the exact value of cos(theta) is √5/3.