If-sin-pi-cos-cos-pi-sin-then-prove-that-sin-2-3-4-

Question Number 13401 by Tinkutara last updated on 19/May/17

If \sin (π \cos θ) = \cos (π \sin θ), then

prove that \sin 2 θ = \pm \frac{3}{4}

Answered by ajfour last updated on 19/May/17

\sin x = \cos y

x = \frac{π}{2} \pm y (a t l e a s t)

h e r e \sin (π \cos θ) = \cos (π \sin θ)

π \cos θ = \frac{π}{2} + π \sin θ \dots \dots (i)

π \cos θ = \frac{π}{2} - π \sin θ \dots \dots (i i)

f r o m (i) :

\cos θ - \sin θ = \frac{1}{2}, s q u a r i n g

1 - \sin 2 θ = \frac{1}{4} \Rightarrow \sin 2 θ = \frac{3}{4};

f r o m (i i) :

\cos θ + \sin θ = \frac{1}{2}, s q u a r i n g

1 + \sin 2 θ = \frac{1}{4} \Rightarrow \sin 2 θ = - \frac{3}{4} .

Commented by Tinkutara last updated on 20/May/17

In (i) {shouldn}^{'} t it be something else

because \cos (\frac{π}{2} + θ) = - \sin θ ?

Commented by ajfour last updated on 20/May/17

i h a v e o b t a i n e d + \frac{3}{4} a s w e l l .

Commented by ajfour last updated on 20/May/17

i f \sin A = \cos B

w e c o n c l u d e A = \frac{π}{2} + B, h e r e i n (i)

Commented by Tinkutara last updated on 20/May/17

Thanks!