How do you solve $sin (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ $sin(x/2)=sqrt2/2$ ?

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How do you solve $sin (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ $sin(x/2)=sqrt2/2$ ?

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Answer 1 · 2016-12-30T13:53:54

$\frac{π}{2} + 4 k π$ $pi/2 + 4kpi$
$\frac{3 π}{4} + 4 k π$ $(3pi)/4 + 4kpi$

Explanation:

$sin (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ $sin (x/2) = sqrt2/2$
Trig table gives:
$\frac{sin x}{2} = \frac{\sqrt{2}}{2}$ $sin x/2 = sqrt2/2$ --> arc $\frac{x}{2} = \frac{π}{4}$ $x/2 = pi/4$ .
Unit circle gives another arc x that has the same sin value:
$\frac{x}{2} = π - \frac{π}{4} = \frac{3 π}{4}$ $x/2 = pi - pi/4 = (3pi)/4$

a. $\frac{x}{2} = \frac{π}{4} + 2 k π$ $x/2 = pi/4 + 2kpi$ -->
$x = \frac{2 π}{4} + 4 k π = \frac{π}{2} + 4 k π$ $x = (2pi)/4 + 4kpi = pi/2 + 4kpi$

b. $\frac{x}{2} = \frac{3 π}{4} + 2 k π$ $x/2 = (3pi)/4 + 2kpi$
$x = \frac{6 π}{4} + 4 k π = \frac{3 π}{2} + 4 k π$ $x = (6pi)/4 + 4kpi = (3pi)/2 + 4kpi$

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How do you solve $sin (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ $sin(x/2)=sqrt2/2$ ?

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How do you solve $sin (\frac{x}{2}) = \frac{\sqrt{2}}{2}$ $sin(x/2)=sqrt2/2$ ?