Given: cot(theta) = 3/5 and sec(theta) < 0cot(θ)=35andsec(θ)<0
Use the identity tan (theta) = 1/cot(theta)tan(θ)=1cot(θ):
tan(theta) = 1/(3/5)tan(θ)=135
tan(theta) = 5/3tan(θ)=53
Use the identity 1 + tan^2(theta) = sec^2(theta)1+tan2(θ)=sec2(θ)
1 + (5/3)^2 = sec^2(theta)1+(53)2=sec2(θ)
sec^2(theta) = 34/9sec2(θ)=349
sec(theta) = -sqrt(34)/3sec(θ)=−√343
Use the identity cos(theta) = 1/sec(theta)cos(θ)=1sec(θ)
cos(theta) = -3/sqrt(34)cos(θ)=−3√34
cos(theta) = -(3sqrt(34))/34cos(θ)=−3√3434
Use the identity tan(theta) = sin(theta)/cos(theta)tan(θ)=sin(θ)cos(θ)
tan(theta)cos(theta) = sin(theta)tan(θ)cos(θ)=sin(θ)
sin(theta) = 5/3(-(3sqrt(34))/34)sin(θ)=53(−3√3434)
sin(theta) = (-5sqrt(34))/34sin(θ)=−5√3434
Use the identity csc(theta) = 1/sin(theta)csc(θ)=1sin(θ)
csc(theta) = 1/(-5/sqrt(34)csc(θ)=1−5√34
csc(theta) = -sqrt(34)/5csc(θ)=−√345
