Given that θ is in Quadrant II and sin(θ) = 2/3, we can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to find the values of the trigonometric functions for θ.
a. To find cos(θ), we can use the identity:
cos^2(θ) = 1 - sin^2(θ)
Plugging in the value of sin(θ) = 2/3:
cos^2(θ) = 1 - (2/3)^2
cos^2(θ) = 1 - 4/9
cos^2(θ) = 5/9
Since θ is in Quadrant II, cos(θ) is negative. Taking the negative square root:
cos(θ) = -√(5/9) = -√5/3
b. To find tan(θ), we can use the identity:
tan(θ) = sin(θ) / cos(θ)
Plugging in the values of sin(θ) and cos(θ):
tan(θ) = (2/3) / (-√5/3)
tan(θ) = -2/√5
c. To find sec(θ), we can use the identity:
sec(θ) = 1 / cos(θ)
Plugging in the value of cos(θ):
sec(θ) = 1 / (-√5/3)
sec(θ) = -3/√5
d. To find csc(θ), we can use the identity:
csc(θ) = 1 / sin(θ)
Plugging in the value of sin(θ):
csc(θ) = 1 / (2/3)
csc(θ) = 3/2
e. To find cot(θ), we can use the identity:
cot(θ) = 1 / tan(θ)
Plugging in the value of tan(θ):
cot(θ) = 1 / (-2/√5)
cot(θ) = -√5/2
Therefore, the exact values of the trigonometric functions for θ in Quadrant II with sin(θ) = 2/3 are:
a. cos(θ) = -√5/3
b. tan(θ) = -2/√5
c. sec(θ) = -3/√5
d. csc(θ) = 3/2
e. cot(θ) = -√5/2