Using Sum-to-Product Formulas Solve the equation by first using a Sum-to-Product Formula.

$\mathrm{sin}\theta +\mathrm{sin}3\theta =0$

Josalynn
2021-08-11
Answered

Using Sum-to-Product Formulas Solve the equation by first using a Sum-to-Product Formula.

$\mathrm{sin}\theta +\mathrm{sin}3\theta =0$

You can still ask an expert for help

hajavaF

Answered 2021-08-12
Author has **90** answers

Approach:

The range of the trigonometric functions of$\mathrm{sin}\theta$ is lie between $[-1,1]$ . No solution exists beyond this range.

Simplify the equation.

Obtain the factors of the equation.

The sum-to-product formulas for cosine is,

$\mathrm{sin}u+\mathrm{sin}v=2\mathrm{sin}\frac{u+v}{2}\mathrm{cos}\frac{u+v}{2}$

Cosine function has period$2\pi$ , thus find the solution in any interval of length $2\pi$ . Sine function is positive in first and second quadrant.

Calculation:

Consider the equation.

$\mathrm{sin}\theta +\mathrm{sin}3\theta =0$

Use Sum-to-Product formulas in the above equation,

$\mathrm{sin}\theta +\mathrm{sin}3\theta =0$

$2\mathrm{sin}\frac{\theta +3\theta}{2}\mathrm{cos}\frac{\theta -3\theta}{2}=0$

$2\mathrm{sin}2\theta \mathrm{cos}\theta =0$

Use the zero product property,

$\mathrm{sin}2\theta =0\dots \left(1\right)$

$\mathrm{cos}\theta =0\dots \left(2\right)$

Consider equation (1).

$\mathrm{sin}2\theta =0$

Taking sine inverse both sides,

${\mathrm{sin}}^{-1}\mathrm{sin}2\theta ={\mathrm{sin}}^{-1}\left(0\right)$

$2\theta ={\mathrm{sin}}^{-1}\left(0\right)$

$2\theta =0,\pi$

The solution of the equation is obtained by adding in the integer multiples of$\pi$ ,

$2\theta =k\pi$

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

Consider equation (2).

$\mathrm{cos}\theta =0$

Taking$\mathrm{cos}}^{-1$ both sides,

${\mathrm{cos}}^{-1}\mathrm{cos}\theta ={\mathrm{cos}}^{-1}\left(0\right)$

$\theta ={\mathrm{cos}}^{-1}\left(0\right)$

$\theta =\frac{\pi}{2}$

The solution of the equation is obtained by adding in the integer multiples of$\pi$ ,

$\theta =\frac{\pi}{2}+k\pi$

The compact general solution is$\theta =\frac{\pi k}{2}$ .

Therefore, the solution of the trigonometry equation$\mathrm{sin}\theta +\mathrm{sin}3\theta =0$ is $\theta =\frac{\pi k}{2}$

The range of the trigonometric functions of

Simplify the equation.

Obtain the factors of the equation.

The sum-to-product formulas for cosine is,

Cosine function has period

Calculation:

Consider the equation.

Use Sum-to-Product formulas in the above equation,

Use the zero product property,

Consider equation (1).

Taking sine inverse both sides,

The solution of the equation is obtained by adding in the integer multiples of

Consider equation (2).

Taking

The solution of the equation is obtained by adding in the integer multiples of

The compact general solution is

Therefore, the solution of the trigonometry equation

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