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If θ is in Quadrant II and sin(θ)= 2/3find the following exactly a. cos(θ) b. tan(θ) c. sec(θ) d. csc(θ) e. cot(θ)

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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

User Dan Healy
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