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Question Number 165170 by Tawa11 last updated on 26/Jan/22

Commented by mr W last updated on 27/Jan/22

Commented by mr W last updated on 27/Jan/22

$g e n e r a l l y :$ $i f k + m = 2 n, t h e n$ $x = \frac{45 °}{n}$
Commented by Tawa11 last updated on 27/Jan/22

$I appreciate sir .$
	Answered by aleks041103 last updated on 27/Jan/22

	Commented by aleks041103 last updated on 27/Jan/22

	$o b v i o u s c o n s t r a i n t :$ $7 x < 90 °$ $\Rightarrow x < 12 \frac{6}{7} °$ $\frac{O C}{O B} = t g (5 x)$ $\frac{O B}{O A} = c t g (3 x)$ $\frac{O A}{O D} = t g (5 x)$ $\frac{O D}{O C} = c t g (7 x)$ $\Rightarrow \frac{{t g}^{2} (5 x)}{t g (3 x) t g (7 x)} = 1$ $\Rightarrow t g (3 x) t g (7 x) = {t g}^{2} (5 x)$ $t g (10 x) = \frac{2 t g (5 x)}{1 - {t g}^{2} (5 x)} = \frac{t g (3 x) + t g (7 x)}{1 - t g (3 x) t g (7 x)}$ $\Rightarrow 2 t g (5 x) = t g (3 x) + t g (7 x) o r t g (3 x) t g (7 x) = {t g}^{2} (5 x) = 1$ $1) 2 t g (5 x) = t g (3 x) + t g (7 x)$ $\Rightarrow 2 t g (5 x) = \frac{s i n (10 x)}{c o s (3 x) c o s (7 x)} = \frac{2 s i n (5 x) c o s (5 x)}{c o s (3 x) c o s (7 x)} = \frac{2 s i n (5 x)}{c o s (5 x)}$ $\Rightarrow {c o s}^{2} (5 x) = c o s (3 x) c o s (7 x)$ $b u t t g (3 x) t g (7 x) = {t g}^{2} (5 x)$ $\Rightarrow {s i n}^{2} (5 x) = s i n (3 x) s i n (7 x)$ $\Rightarrow 1 = {c o s}^{2} (5 x) + {s i n}^{2} (5 x) =$ $= c o s (3 x) c o s (7 x) + s i n (3 x) s i n (7 x) =$ $= c o s (4 x) = 1$ $\Rightarrow 4 x = k 360 °$ $\Rightarrow x = k . 90 °$ $b u t x < 13 ° \Rightarrow k = 0 \Rightarrow x = 0 \to t r i v i a l$ $2) {t g}^{2} (5 x) = t g (3 x) t g (7 x) = 1$ $\Rightarrow t g (5 x) = \pm 1$ $5 x = 45 °, 135 °, 225 °, 315 °$ $\Rightarrow x = 9 °, 27 °, 45 °, 63 °$ $b u t x < 13 ° \Rightarrow x = 9 ° o n l y p o s s i b l e$ $c h e c k :$ $t g (27 °) t g (63 °) = t g (27 °) c t g (90 - 63) =$ $= t g (27 °) c t g (27 °) = 1 ✓$ $\Rightarrow x = 9 ° = \frac{π}{20} r a d$
	Commented by Tawa11 last updated on 27/Jan/22

	$I appreciate sir$
	Answered by nurtani last updated on 27/Jan/22

	$7 x + 5 x + 5 x + 3 x = 180 °$ $20 x = 180 °$ $x = \frac{180 °}{20} = 9 °$
	Commented by mr W last updated on 27/Jan/22

	$i t h i n k {i t}^{'} s j u s t b y a c c i d e n t t r u e .$ $t h e m e t h o d {c a n}^{'} t b e g e n e r a l l y a p p l i e d,$ $f o r e x a m p l e i n f o l l o w i n g c a s e :$
	Commented by mr W last updated on 27/Jan/22

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