sin-sin-sin-2-Express-explicitly-in-terms-of-

Question Number 39486 by ajfour last updated on 06/Jul/18

\sin θ = \sin α \sin (\frac{θ + α}{2})

E x p r e s s θ e x p l i c i t l y i n t e r m s o f α .

Commented by math khazana by abdo last updated on 07/Jul/18

\Rightarrow 2 s i n (\frac{θ}{2}) c o s (\frac{θ}{2}) = s i n α (s i n (\frac{θ}{2}) c o s (\frac{α}{2}) +

c o s (\frac{θ}{2}) s i n (\frac{α}{2})) l e t p u t s i n (\frac{θ}{2}) = x \Rightarrow

2 x ξ \sqrt{1 - x^{2}} = s i n α (x c o s (\frac{α}{2}) + ξ \sqrt{1 - x^{2}} s i n (\frac{α}{2})) \Rightarrow

2 x ξ \sqrt{1 - x^{2}} = s i n α c o s (\frac{α}{2}) x + s i n α s i n (\frac{α}{2}) ξ \sqrt{1 - x^{2}}

(2 x ξ - s i n (α) s i n (\frac{α}{2}) ξ) \sqrt{1 - x^{2}} = s i n (α) c o s (\frac{α}{2}) x \Rightarrow

{(2 x - s i n (α) s i n (\frac{α}{2}))}^{2} (1 - x^{2}) = {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2}) x^{2}

4 x^{2} - 4 s i n (α) s i n (\frac{α}{2}) x + {s i n}^{2} (α) {s i n}^{2} (\frac{α}{2})

- {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2}) x^{2} = 0 \Rightarrow

(4 - {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2})) x^{2} - 4 s i n (α) s i n (\frac{α}{2}) x

+ {s i n}^{2} (α) {s i n}^{2} (\frac{α}{2}) = 0

Δ^{^{'}} = 4 {s i n}^{2} (α) {s i n}^{2} (\frac{α}{2}) - (4 - {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2})) ({s i n}^{2} α {s i n}^{2} (\frac{α}{2}))

= 4 {s i n}^{2} (α) {s i n}^{2} (\frac{α}{2}) - 4 {s i n}^{2} (α) {s i n}^{2} (\frac{α}{2})

+ {s i n}^{4} (α) {s i n}^{2} (\frac{α}{2}) {c o s}^{2} (\frac{α}{2})

= \frac{1}{4} {s i n}^{6} (α) \Rightarrow x = \frac{2 s i n (α) s i n (\frac{α}{2}) \bar{+} \frac{1}{2} ∣ s i n α ∣^{3}}{4 - {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2})} \Rightarrow

θ = 2 a r c s i n (x) \Rightarrow

θ = 2 a r c s i n {\frac{2 s i n (α) s i n (\frac{α}{2}) \bar{+} \frac{1}{2} ∣ s i n α ∣^{3}}{4 - {s i n}^{2} (α) {c o s}^{2} (\frac{α}{2})}} .

Commented by tanmay.chaudhury50@gmail.com last updated on 07/Jul/18

e x c e l l e n t q u e s t i o n . . p l s g i v e t i m e \dots

Commented by math khazana by abdo last updated on 07/Jul/18

ξ^{2} = 1

Commented by ajfour last updated on 07/Jul/18

Commented by ajfour last updated on 07/Jul/18

I f w e t a k e ∠ C A D t o b e θ a n d

e x p r e s s θ i n t e r m s o f α . W e t h e n

o b t a i n a i n t e r m s o f r a d i u s R

a n d α (w h i c h w a s t h e q u e s t i o n) .

Commented by ajfour last updated on 07/Jul/18

T h a n k y o u S i r, l e t m e s e e i f i c a n

f o l l o w y o u r s o l u t i o n .

Commented by ajfour last updated on 08/Jul/18

l i n e 6 t o l i n e 7 h o w S i r, p l e a s e c h e c k;

w h a t a b o u t t e r m w i t h x^{4} a n d x^{3} ?

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