Solve-for-real-numbers-x-2-2x-3siny-4cosy-6-0-

Question Number 149478 by mathdanisur last updated on 05/Aug/21

S o l v e f o r r e a l n u m b e r s :

x^{2} - 2 x - 3 s i n \boldsymbol y - 4 c o s \boldsymbol y + 6 = 0

Commented by iloveisrael last updated on 06/Aug/21

Δ ⩾ 0

3 \sin y + 4 cosy - 6 ⩾ - 1

5 \sin (y + \tan^{- 1} (\frac{4}{3})) ⩾ 5

\sin (y + \tan^{- 1} (\frac{4}{3})) = 1

\Rightarrow y + \tan^{- 1} (\frac{4}{3}) = \frac{π}{2} + 2 k π

\Rightarrow y = \frac{π}{2} - \tan^{- 1} (\frac{4}{3}) + 2 k π

Commented by mathdanisur last updated on 06/Aug/21

Thankyou \boldsymbol Ser

Answered by mr W last updated on 05/Aug/21

3 s i n \boldsymbol y - 4 c o s \boldsymbol y = x^{2} - 2 x + 6

5 (s i n \boldsymbol y \frac{3}{5} - c o s \boldsymbol y \frac{4}{5}) = x^{2} - 2 x + 6

5 (s i n \boldsymbol y \cos α - c o s \boldsymbol y \sin α) = {(x - 1)}^{2} + 5

w i t h \cos α = \frac{3}{5}, \sin α = \frac{4}{5}, \tan α = \frac{4}{3}

5 \sin (y - α) = {(x - 1)}^{2} + 5

5 \sin (y - \tan^{- 1} \frac{4}{3}) = {(x - 1)}^{2} + 5

L H S ⩽ 5

R H S ⩾ 5

\Rightarrow L H S = R H S = 5

\Rightarrow x = 1

\Rightarrow \sin (y - \tan^{- 1} \frac{4}{3}) = 1

\Rightarrow y - \tan^{- 1} \frac{4}{3} = 2 k π + \frac{π}{2}

\Rightarrow y = 2 k π + \frac{π}{2} + \tan^{- 1} \frac{4}{3}

Commented by mathdanisur last updated on 05/Aug/21

Thank you \boldsymbol Ser, cool

Commented by peter frank last updated on 05/Aug/21

e x p l a n a t i o n 2 n d a n d 3 r d l i n e

Commented by mr W last updated on 05/Aug/21

i a d d e d s o m e l i n e s m o r e .

Commented by peter frank last updated on 05/Aug/21

t h a n k y o u

Commented by Tawa11 last updated on 05/Aug/21

great

Answered by MJS_new last updated on 05/Aug/21

x = 1 \pm \sqrt{- 5 + 3 \sin y + 4 \cos y}

- 10 ⩽ - 5 + 3 \sin y + 4 \cos y ⩽ 0

\Rightarrow

- 5 + 3 \sin y + 4 \cos y = 0 \Leftrightarrow y = 2 n π + \arctan \frac{3}{4}

\Rightarrow

solution is

x = 1 \land y = 2 n π + \arctan \frac{3}{4} \forall n \in Z

Commented by mathdanisur last updated on 05/Aug/21

Thank you Ser, cool

Thank you \boldsymbol Ser, cool