Mid-Chapter Quiz
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Consider using the Pythagorean Identity cot ^2 θ +1= csc ^2 θ .
See solution.
We want to find the exact value of csc θ. Let's assume that cot θ = - 12 because the value of cot θ cannot be positive when 270^(∘)<θ<360^(∘). To find cscθ, we will use one of the Pythagorean Identities.
cot ^2 θ +1= csc ^2 θ
Let's do it!
LHS^2=RHS^2
LHS+1=RHS+1
cot ^2 θ +1= csc ^2 θ
sqrt(LHS)=sqrt(RHS)
sqrt(a/b)=sqrt(a)/sqrt(b)
Calculate root
Be aware that we are told that θ lies between 270^(∘) and 360^(∘). Therefore, θ is in Quadrant IV.
In this quadrant, the sine of θ is negative. Since csc θ= 1sin θ, the sign of csc θ is also negative in this quadrant. Therefore, we will only keep the negative solution. csc θ =- sqrt(5)/2