How do you solve $csc (theta/2) = sin (theta/2)$ ?

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Answer 1 · 2015-05-21T16:32:26

$csc (x/2) = 1/(sin (x/2)) = sin (x/2)$

$sin^2 (x/2) = 1 -> sin (x/2) = +- 1$

a. $sin (x/2) = 1 = sin (pi/2) -> x/2 = pi/2 -> x = pi$

b. $sin (x/2) = -1 = sin ((3pi)/2) -> x/2 = (3pi)/2 -> x = 3pi$

ChecK:

$x = pi -> x/2 = pi/2 --> sin (pi)/2 = 1 -> csc x = 1$ OK
$x = (3pi)/2$ -> $x/2 = (3pi)/6 = sin ((pi)/2) = 1 -> csc = 1$ OK

How do you solve csc (theta/2) = sin (theta/2)csc (theta/2) = sin (theta/2)?