How do you find $sin(x/2)$ if $cscx=3$ ?

Question

10272 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-05-17T16:47:09

Since csc x=3, sin x= $1/3$ and therefore cos x= $sqrt(8/9)$ = $(2sqrt2)/3$

Since cosx equals $1- 2sin^2 (x/2)$ , hence

$2sin^2 (x/2)= 1-cosx$

= 1- $(2sqrt2)/3$ = $(3-2sqrt2)/3$ = $(sqrt2 -1)^2 /3$

$sin^2 (x/2)= (sqrt2 -1)^2 /6$

$sin(x/2) = (sqrt2-1)/sqrt6$

Answer 2 · 2015-05-17T16:57:45

Use trig identity: $1 + cot^2 x = 1/sin^2 x = csc^2 x$

$1/sin^2 x = 9 -> sin^2 x = 1/9 -> sin x = +1/3$

$cos x = 1 - sin^2 x = 1 - 1/9 = 8/9 -> cos x = +(sqrt2)/3$

To find $sin (x/2)$ use trig identity: $2sin^2 (x/2) = 1 - cos x.$

$2sin^2 (x/2) = 1 - cos x = 3/3 - (sqrt2)/3 = 0.53$

$sin^2 (x/2) = 0.265 -> sin (x/2) = 0.51$

How do you find sin(x/2)sin(x/2) if cscx=3cscx=3?