Question

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

Factors and multiples
Use a Double or Half-Angled Formula to solve the equation in the interval $$\displaystyle{\left[{0},{2}\pi\right)}$$.
$$\displaystyle{\sin{\theta}}-{\cos{\theta}}={\frac{{{1}}}{{{2}}}}$$

2021-08-17

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,
$$\displaystyle{\sin{{2}}}{x}={2}{\sin{{x}}}{\cos{{x}}}$$
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,
$$\displaystyle{\left({\sin{{t}}}{h}{e}{t}-{\cos{\theta}}\right)}^{{{2}}}={\left({\frac{{{1}}}{{{2}}}}\right)}^{{{2}}}$$
$$\displaystyle{{\cos}^{{{2}}}\theta}+{{\sin}^{{{2}}}\theta}-{2}{\cos{\theta}}{\sin{\theta}}={\frac{{{1}}}{{{4}}}}$$
$$\displaystyle{1}-{2}{\cos{\theta}}{\sin{\theta}}={\frac{{{1}}}{{{4}}}}$$
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 $$\displaystyle{\left[{0},{2}\pi\right)}$$ 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 $$\displaystyle{\left[{0},{2}\pi\right)}$$ is $$\theta=1.15, 3.56$$.