Solve the equation by first using a Sum-to-Product Formula. \sin \theta + \sin 3 \theta = 0

Josalynn 2021-08-11 Answered
Using Sum-to-Product Formulas Solve the equation by first using a Sum-to-Product Formula.
\(\displaystyle{\sin{\theta}}+{\sin{{3}}}\theta={0}\)

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Expert Answer

hajavaF
Answered 2021-08-12 Author has 14809 answers
Approach:
The range of the trigonometric functions of \(\displaystyle{\sin{\theta}}\) is lie between \(\displaystyle{\left[-{1},{1}\right]}\). No solution exists beyond this range.
Simplify the equation.
Obtain the factors of the equation.
The sum-to-product formulas for cosine is,
\(\displaystyle{\sin{{u}}}+{\sin{{v}}}={2}{\sin{{\frac{{{u}+{v}}}{{{2}}}}}}{\cos{{\frac{{{u}+{v}}}{{{2}}}}}}\)
Cosine function has period \(\displaystyle{2}\pi\), thus find the solution in any interval of length \(\displaystyle{2}\pi\). Sine function is positive in first and second quadrant.
Calculation:
Consider the equation.
\(\displaystyle{\sin{\theta}}+{\sin{{3}}}\theta={0}\)
Use Sum-to-Product formulas in the above equation,
\(\displaystyle{\sin{\theta}}+{\sin{{3}}}\theta={0}\)
\(\displaystyle{2}{\sin{{\frac{{\theta+{3}\theta}}{{{2}}}}}}{\cos{{\frac{{\theta-{3}\theta}}{{{2}}}}}}={0}\)
\(\displaystyle{2}{\sin{{2}}}\theta{\cos{\theta}}={0}\)
Use the zero product property,
\(\displaystyle{\sin{{2}}}\theta={0}\ldots{\left({1}\right)}\)
\(\displaystyle{\cos{\theta}}={0}\ldots{\left({2}\right)}\)
Consider equation (1).
\(\displaystyle{\sin{{2}}}\theta={0}\)
Taking sine inverse both sides,
\(\displaystyle{{\sin}^{{-{1}}}{\sin{{2}}}}\theta={{\sin}^{{-{1}}}{\left({0}\right)}}\)
\(\displaystyle{2}\theta={{\sin}^{{-{1}}}{\left({0}\right)}}\)
\(\displaystyle{2}\theta={0},\pi\)
The solution of the equation is obtained by adding in the integer multiples of \(\displaystyle\pi\),
\(\displaystyle{2}\theta={k}\pi\)
\(\displaystyle\theta={\frac{{{k}\pi}}{{{2}}}}\)
Consider equation (2).
\(\displaystyle{\cos{\theta}}={0}\)
Taking \(\displaystyle{{\cos}^{{-{1}}}}\) both sides,
\(\displaystyle{{\cos}^{{-{1}}}{\cos{\theta}}}={{\cos}^{{-{1}}}{\left({0}\right)}}\)
\(\displaystyle\theta={{\cos}^{{-{1}}}{\left({0}\right)}}\)
\(\displaystyle\theta={\frac{{\pi}}{{{2}}}}\)
The solution of the equation is obtained by adding in the integer multiples of \(\displaystyle\pi\),
\(\displaystyle\theta={\frac{{\pi}}{{{2}}}}+{k}\pi\)
The compact general solution is \(\displaystyle\theta={\frac{{\pi{k}}}{{{2}}}}\).
Therefore, the solution of the trigonometry equation \(\displaystyle{\sin{\theta}}+{\sin{{3}}}\theta={0}\) is \(\displaystyle\theta={\frac{{\pi{k}}}{{{2}}}}\)
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