Find-y-in-3rd-quadrant-tan-y-30-cot-y-

Question Number 21241 by tawa tawa last updated on 17/Sep/17

Find y in 3 rd quadrant

\tan (y - 30) = \cot (y)

Answered by sma3l2996 last updated on 17/Sep/17

t a n (y - 30) = c o t (y)

t a n (y - 30) = \frac{1}{t a n (y)} \Leftrightarrow \frac{t a n (y) - t a n (30)}{1 + t a n (y) t a n (30)} = \frac{1}{t a n y}

w i t h t a n (30) = \frac{\sqrt{3}}{3}

t a n (y) (t a n (y) - t a n (30)) = 1 + t a n (y) t a n (30)

{t a n}^{2} (y) - \frac{\sqrt{3}}{3} t a n (y) - \frac{\sqrt{3}}{3} t a n (y) = 1

{t a n}^{2} (y) - 2 \frac{\sqrt{3}}{3} t a n (y) - 1 = 0

Δ = \frac{4}{3} + 4 = \frac{16}{3} = {(\frac{4 \sqrt{3}}{3})}^{2}

t a n (y) = \frac{\frac{2 \sqrt{3}}{3} - \frac{4 \sqrt{3}}{3}}{2} = \frac{- \sqrt{3}}{3} \Leftrightarrow y = - \frac{π}{6} + k π

o r t a n (y) = \frac{\frac{2 \sqrt{3}}{3} + \frac{4 \sqrt{3}}{3}}{2} = \sqrt{3} \Leftrightarrow y = \frac{π}{3} + k π

Commented by tawa tawa last updated on 17/Sep/17

God bless you sir .