Is tan 1° rational or irrational? Give your proof.


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Question Number 54923 by mr W last updated on 14/Feb/19

$I s \tan 1 ° r a t i o n a l o r i r r a t i o n a l ?$ $G i v e y o u r p r o o f .$
Commented by Otchere Abdullai last updated on 14/Feb/19

$t a n 1 ° = 0.01745506493$ $t h u s t a n 1 ° i s i r r a t i o n a l$ $R e a s o n : T h i s i s b e c a u s e t a n 1 ° h a s$ $n o n - r e p e a t i n g p a t t e r n d e c i m a l$ $n u m b e r s$
Commented by mr W last updated on 14/Feb/19

$h o w c a n y o u b e s u r e ? m a y b e i t s d e c i m a l$ $r e p e a t s j u s t a f t e r o n e m i l l i o n d i g i t s .$ $s o r r y, w e n e e d a n e x a c t p r o o f .$
Commented by Otchere Abdullai last updated on 14/Feb/19

$o k p r o f$
Commented by Otchere Abdullai last updated on 15/Feb/19

$s t e p 1 . A s s u m e t h a t t a n (1) i s r a t i o n a l$ $2 . t h e r e f o r e t a n (2) i s r a t i o n a l$ $3 . I n g e n e r a l b y i n d u c t i o n t a n (k) i s$ $r a t i o n a l$ $4 . T a n (30) i s r a t i o n a l$ $5 . c o n t r a d i c t i o n$ $t a n (1 + 1) = \frac{t a n (1) + t a n (1)}{1 - {t a n}^{2} (1)} \in Q$ $t a n (30) = \frac{1}{\sqrt{3}} \notin Q$ $t a n (k + 1) = \frac{t a n (k) + t a n (1)}{1 - t a n (k) t a n (1)} \in Q$ $∴ t a n 1 ° i s i r r a t i o n a l$
Commented by mr W last updated on 15/Feb/19

$t h a n k y o u s i r! g o o d a p p r o a c h!$
Commented by Otchere Abdullai last updated on 15/Feb/19

$M o s t w e l c o m e p r o f W$
Commented by $@ty@m last updated on 15/Feb/19

$(1) I f t a n 1 i s r a t i o n a l t h e n h o w c a n y o u$ $c o n c l u d e t h a t \tan 2 i s a l s o r a t i o n a l ?$ $(2) F u r t h e r i f y o u r e a c h \tan 45$ $b y i n d u c t i o n {y o u}^{'} d g e t a r a t i o n a l$ $n u m b e r .$
Commented by mr W last updated on 16/Feb/19

$i f \tan 1 ° i s r a t i o n a l, s a y \tan 1 ° = \frac{q}{p}$ $\tan 2 ° = \frac{2 \times \frac{q}{p}}{1 - \frac{q^{2}}{p^{2}}} = \frac{2 p q}{(p - q) (p + q)} = \frac{n}{m}$ $i . e . i f \tan 1 ° i s r a t i o n a l, t h e n$ $\Rightarrow \tan 2 ° i s a l s o r a t i o n a l$ $\Rightarrow \tan k ° i s a l s o r a t i o n a l$ $\Rightarrow \tan 45 ° i s a l s o r a t i o n a l (t h i s i s t r u e)$ $\Rightarrow \tan 60 ° i s a l s o r a t i o n a l (b u t t h i s i s n o t t r u e)$ $s i n c e \tan k ° i s n o t a l w a y s r a t i o n a l,$ $i t i s n o t t r u e t h a t \tan 1 ° i s r a t i o n a l,$ $i n o t h e r w o r d s \tan 1 ° m u s t b e i r r a t i o n a l .$
Commented by $@ty@m last updated on 17/Feb/19

$T h a n k s f o r c l a r i f i c a t i o n .$
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