If sin(θ)=2/7, and θ is in quadrant II, find cos(θ), sec(θ), csc(θ), tan(θ), cot(θ).

Question

If sin(θ)=2/7, and θ is in quadrant II, find cos(θ), sec(θ), csc(θ), tan(θ), cot(θ).

Answer 1

sinT=2/7

cscT=1/sinT= 7/2

all other trig funtions are negative in quadrant 2

cosT = =-sqr(7^2-2^2)= -sqr(49-4)/7=-sqr45/7 = -(3/7)sqr5

secT=1/cosT=-7/3sqr5

tanT=sinT/cosT= -2/3sqr5

cotT=1/tanT= -3sqr5/2

Answer 2

Only sin and csc are positive in the 2nd quadrant.

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sin = opposite/hypotenuse = 2/7

opposite = 2, hypotenuse = 7

2^2 + x^2 = 7^2

x^2 = 49-4 = 45

x = sqrt(45) = sqrt(9*5) = 3*sqrt(5) = adjacent side

csc = hyptoe/15nuse/opposite = 7/2 = 4.5

The rest of these are negative in the 2nd quadrant. I just made the sign negative on the last step.

cos = adj/hyp = - 3sqrt(5)/7

sec = 7/3sqrt(5) if you have to rationalize the denominator = 7/3sqrt(5) * sqrt(5)/sqrt(5) = -7sqrt(5)/15

tan = opposite / adjacent = 2/3sqrt5 rationalize denominator: 2/3sqrt(5) * sqrt(5)/sqrt(5) = -2sqrt(5)/15

cot = adjacent / opposite = -3sqrt(5)/2