Question

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

Factors and multiples
ANSWERED
asked 2021-08-16
Use a Double or Half-Angled Formula to solve the equation in the interval \(\displaystyle{\left[{0},{2}\pi\right)}\).
\(\displaystyle{\sin{\theta}}-{\cos{\theta}}={\frac{{{1}}}{{{2}}}}\)

Expert Answers (1)

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 \( \theta=1.15, 3.56\).

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