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

DofotheroU

DofotheroU

Answered question

2021-08-16

Use a Double or Half-Angled Formula to solve the equation in the interval [0,2π).
sinθcosθ=12

Answer & Explanation

unett

unett

Skilled2021-08-17Added 119 answers

Approach:
The range of the trigonometric functions of sinθ and cosθ 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,
sin2x=2sinxcosx
Cosine and sine functions has period of 2π, thus find the solution in any interval of length 2π
Cosine function is positive in first and fourth quadrant. SIne function is positive in first and second quadrant.
Calculation:
Consider the equation.
sinθcosθ=12
Squaring both sides in above equation,
(sinthetcosθ)2=(12)2
cos2θ+sin2θ2cosθsinθ=14
12cosθsinθ=14
Use doubled-Angled the equation,
sin2θ=34
Taking sine inverse both sides,
sin1sin2θ=sin1(34)
2θ=sin1(34)
2θ0.85
θ0.43
The solution of the equation is obtained by adding in the integer multiples of π,
θ0.43+kπ
Substitute k=0,1,2
θ0.43+0=0.43
θ0.43+π=3.56
θ0.43+2π=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 sinθcosθ=12 in the interval [0,2π) is θ=1.15,3.56.
Conclusion:
Hence, the solution of the trigonometry equation sinθcosθ=12 in the interval [0,2π) is θ=1.15,3.56.

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