In a right triangle:

tan A = opp/adj

opp/adj = -2/3

opp = 2

adj = 3 (ignore the sign for now)

The angle lies in quadrant II where sin > 0, cos < 0, tan < 0

hyp = sqrt( 2^2+ 3^2) = sqrt(4+9) = sqrt(13)

sin A = opp/hyp = 2/sqrt(13) ( sine is positive)

cos A = adj/hyp = -3/sqrt(13) (cos is negative)

tan A = opp/adj = -2/3 (tan is negative)

cot A = 1/tan A = -3/2

sec A = 1/cos A = -sqrt(13) /3

csc A = 1/sin A = sqrt(13)/2

tan A = - 2/3 and A in second quadrant

r² = 2² + 3² = 13

r = √13

sin A = 2 / √13

cos A = - 3 / √13

Assuming a right triangle we have:

opp² + adj² = hyp²

i.e. 2² + 3² = 13 => hyp²

so, hyp = √13

Now, sinA = opp/hyp => 3/√13...positive as sinA > 0 in quadrant 2

And, cosA = adj/hyp => -3/√13..negative as cosA < 0 in quadrant 2

Also, secA => 1/cosA = -√13/3

and cosecA => 1/sinA = √13/3

and cotA => 1/tanA = -3/2

:)>

Hint:

sinA>0

cosA<0

tanA=sinA/cosA

and

sec^2(A)=1+tan^2(A)

do your own homework

sec(a)^2 - tan(a)^2 = 1

sec(a)^2 - (-2/3)^2 = 1

sec(a)^2 - 4/9 = 1

sec(a)^2 = 13/9

sec(a) = +/- sqrt(13) / 3

a is in Q2, so sec(a) < 0

sec(a) = -sqrt(13) / 3

cos(a) = -3 / sqrt(13) = -3 * sqrt(13) / 13

cot(a) = -3/2

csc(a)^2 - cot(a)^2 = 1

csc(a)^2 - (-3/2)^2 = 1

csc(a)^2 - 9/4 = 1

csc(a)^2 = 13/4

csc(a) = +/- sqrt(13) / 2

csc(a) > 0

csc(a) = sqrt(13) / 2

sin(a) = 2/sqrt(13) = 2 * sqrt(13) / 13