How do you find all solutions of the equation $sin (x + \frac{π}{4}) - sin (x - \frac{π}{4}) = 1$ $sin(x+pi/4)-sin(x-pi/4)=1$ in the interval $[0, 2 π)$ $[0,2pi)$ ?

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Answer 1 · 2017-06-15T14:18:50

$"The Soln. Set.="{2kpi+-pi/4 | k in ZZ.}.$

The Solns. in $[0,2pi), are, therefore, pi/4, and, 7pi/4.$

$sin(x+pi/4)-sin(x-pi/4)=1.$

$rArr 2cos[1/2{(x+pi/4)+(x-pi/4)}]sin[1/2{(x+pi/4)-(x-pi/4)}]=1.$

$rArr 2cosxsin(pi/4)=1.$

$rArr2*1/sqrt2*cosx=1.$

$rArr cosx=1/sqrt2=cos(pi/4).$

Knowing that, $costheta=cosalpha rArr theta=2kpi+-alpha, k in ZZ,$

$"The Soln. Set.="{2kpi+-pi/4 | k in ZZ.}.$

In Particular, the Solns. in $[0,2pi), are, therefore, pi/4, and, 7pi/4.$

How do you find all solutions of the equation sin(x+pi/4)-sin(x-pi/4)=1sin(x+π4)−sin(x−π4)=1sin(x+pi/4)-sin(x-pi/4)=1 in the interval [0,2pi)[0,2π)[0,2pi)?