Use a Double or Half-Angled Formula to solve the equation

Approach:
The range of the trigonometric functions of ${\displaystyle \sin \theta }$ and ${\displaystyle \cos \theta }$ are lie between [-1,1]. No solution exists beyond this range.
Simplify the equation.
Obtain the factors of the equation.
Use sine Double-Angled formulas,
${\displaystyle \sin 2x=2\sin x\cos x}$
Cosine and sine functions has period of ${\displaystyle 2\pi }$, thus find the solution in any interval of length ${\displaystyle 2\pi }$
Cosine function is positive in first and fourth quadrant. SIne function is positive in first and second quadrant.
Calculation:
Consider the equation.
${\displaystyle \sin \theta -\cos \theta =\frac{1}{2}}$
Squaring both sides in above equation,
${\displaystyle {(\sin thet-\cos \theta )}^2={\left(\frac{1}{2}\right)}^2}$
${\displaystyle {\cos }^2\theta +{\sin }^2\theta -2\cos \theta \sin \theta =\frac{1}{4}}$
${\displaystyle 1-2\cos \theta \sin \theta =\frac{1}{4}}$
Use doubled-Angled the equation,
${\displaystyle \sin 2\theta =\frac{3}{4}}$
Taking sine inverse both sides,
${\displaystyle {\sin }^{-1}\sin 2\theta ={\sin }^{-1}\left(\frac{3}{4}\right)}$
${\displaystyle 2\theta ={\sin }^{-1}\left(\frac{3}{4}\right)}$
${\displaystyle 2\theta \approx 0.85}$
${\displaystyle \theta \approx 0.43}$
The solution of the equation is obtained by adding in the integer multiples of ${\displaystyle \pi }$,
${\displaystyle \theta \approx 0.43+k\pi }$
Substitute ${\displaystyle k=0,1,2}$
${\displaystyle \theta \approx 0.43+0=0.43}$
${\displaystyle \theta \approx 0.43+\pi =3.56}$
${\displaystyle \theta \approx 0.43+2\pi =1.15}$
From the above obtained solutions only 1.15 and is 3.56 satisfying the equation, so these are the only solutions of the equation.
Therefore, the solution of the trigonometry equation ${\displaystyle \sin \theta -\cos \theta =\frac{1}{2}}$ in the interval ${\displaystyle [0,2\pi )}$ is ${\displaystyle \theta =1.15,3.56}$.
Conclusion:
Hence, the solution of the trigonometry equation ${\displaystyle \sin \theta -\cos \theta =\frac{1}{2}}$ in the interval ${\displaystyle [0,2\pi )}$ is $\theta =1.15,3.56$.