Find the exact values of the other trigonometric functions of θ given that?

Using the unit circle we know that sinϴ<0 in quadrants III and IV.

and cotϴ is a positive value which means that ϴ lies in quadrant III because all tan/cot values are positive in quadrant III.

cotϴ = 7 which means that the tanϴ = 1/7

tan=sin/cos drawing a triangle in QIII and labeling the opposite side -1 and the adjacent -7 (because both sine and cosine are negative in QIII) the hypotenuse would be equal to 5√2 because (-1)²+(-7)²=C² so C=√50=√(2·25)=5√2

now that you have your triangle ready cosϴ= -1/(5√2) sinϴ= -7/(5√2) secϴ=-(5√2) and cscϴ=-(5√2)/7

hyp length=sqrt(49+1)=sqrt(50)

the angle is in the third quadrant.

sin θ=-1/sqrt(50)

cos θ=-7/sqrt(50)

tan θ =1/7

sec θ= -sqrt(50)/ 7

cosec θ= -sqrt(50)