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.
sinθ+sin3θ=0
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Expert Answer

hajavaF
Answered 2021-08-12 Author has 90 answers
Approach:
The range of the trigonometric functions of sinθ 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,
sinu+sinv=2sinu+v2cosu+v2
Cosine function has period 2π, thus find the solution in any interval of length 2π. Sine function is positive in first and second quadrant.
Calculation:
Consider the equation.
sinθ+sin3θ=0
Use Sum-to-Product formulas in the above equation,
sinθ+sin3θ=0
2sinθ+3θ2cosθ3θ2=0
2sin2θcosθ=0
Use the zero product property,
sin2θ=0(1)
cosθ=0(2)
Consider equation (1).
sin2θ=0
Taking sine inverse both sides,
sin1sin2θ=sin1(0)
2θ=sin1(0)
2θ=0,π
The solution of the equation is obtained by adding in the integer multiples of π,
2θ=kπ
θ=kπ2
Consider equation (2).
cosθ=0
Taking cos1 both sides,
cos1cosθ=cos1(0)
θ=cos1(0)
θ=π2
The solution of the equation is obtained by adding in the integer multiples of π,
θ=π2+kπ
The compact general solution is θ=πk2.
Therefore, the solution of the trigonometry equation sinθ+sin3θ=0 is θ=πk2
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