Question

How do you find the exact value of cot θ if csc θ = -3/2 and 180 < θ < 270?

Answer 1

`\displaystyle\\\text{If }~~180^o<\theta<270^o~~\text{then }~~\theta\in~\text{quadrant 3}\\\\\text{In the 3rd cotangent dial is positive.}\\\\\text{We use the formula: } ~~~\boxed{1+\cot^2\theta=\csc^2\theta}`

`\displaystyle\\1+\cot^2\theta=\csc^2\theta\\\\\cot^2\theta=\csc^2\theta-1\\\\\cot^2\theta=\left(-(3)/(2)\right)^2-1\\\\\\\cot^2\theta=\left((3)/(2)\right)^2-1\\\\\\\cot^2\theta=(9)/(4)-(4)/(4)\\\\\\\cot^2\theta=(5)/(4)\\\\\\\cot\theta=\pm\sqrt{(5)/(4)}\\\\\\\text{We will eliminate the negative solution.}\\\\\\\cot\theta=+\sqrt{(5)/(4)}\\\\\\\boxed{\bf\cot\theta=(√(5))/(2)}}`