What is the exact value of the other primary trigonometric ratios of tanTheta=(-5)/12 in quadrant II?

1 Answer
Apr 7, 2018

cot(theta) = -12/5

sec(theta) = -13/12

cos(theta) = -12/13

sin(theta) = 5/13

csc(theta) =13/5

Explanation:

Use the identity cot(theta) = 1/tan(theta):

cot(theta) = -12/5

Use the identity:

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

Substitute tan^2(theta) = (-5/12)^2 :

1+(-5/12)^2 = sec^2(theta)

144/144+25/144

169/144 = sec^2(theta)

sec(theta) = +-13/12

We know that the secant is negative in the second quadrant, therefore, we choose the negative value:

sec(theta) = -13/12

Use the identity cos(theta) = 1/sec(theta):

cos(theta) = -12/13

Use the identity:

tan(theta)= sin(theta)/cos(theta)

Multiply both sides by cos(theta):

sin(theta) = tan(theta)cos(theta)

Substitute tan(theta) = -5/12 and cos(theta) = -12/13:

sin(theta) = (-5/12)(-12/13)

sin(theta) = 5/13

Use the identity csc(theta) = 1/sin(theta):

csc(theta) = 13/5