Start with the identity cos^2(theta) + sin^2(theta) = 1cos2(θ)+sin2(θ)=1
Divide both sides by cos^2(theta)cos2(θ):
cos^2(theta)/cos^2(theta) + sin^2(theta)/cos^2(theta) = 1/cos^2(theta)cos2(θ)cos2(θ)+sin2(θ)cos2(θ)=1cos2(θ)
Use the identity sin(theta)/cos(theta) = tan(theta)sin(θ)cos(θ)=tan(θ) and cos^2(theta)/cos^2(theta) = 1cos2(θ)cos2(θ)=1:
1 + tan^2(theta) = 1/cos^2(theta)1+tan2(θ)=1cos2(θ)
Solve for cos(theta)cos(θ):
cos^2(theta) = 1/(1 + tan^2(theta))cos2(θ)=11+tan2(θ)
cos(theta) = +-sqrt(1/(1 + tan^2(theta)))cos(θ)=±√11+tan2(θ)
The domain restriction pi < theta < (3pi)/2π<θ<3π2 tells us that thetaθ is in the third quadrant (where the cosine function is negative), therefore, we change the +-± to - only:
cos(theta) = -sqrt(1/(1 + tan^2(theta)))cos(θ)=−√11+tan2(θ)
Substitute (2/3)^2(23)2 for tan^2(theta)tan2(θ)
cos(theta) = -sqrt(1/(1 + (2/3)^2))cos(θ)=−
⎷11+(23)2
cos(theta) = -sqrt(1/(1 + 4/9))cos(θ)=−√11+49
cos(theta) = -sqrt(1/(13/9))cos(θ)=−√1139
cos(theta) = -sqrt(9/13)cos(θ)=−√913
cos(theta) = -(3sqrt(13))/13cos(θ)=−3√1313
