 # 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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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 $$\displaystyle\theta={\frac{{\pi{k}}}{{{2}}}}$$