find-the-general-solution-for-sin5-sin3-1-

Question Number 63693 by Rio Michael last updated on 07/Jul/19

f i n d t h e g e n e r a l s o l u t i o n f o r

s i n 5 θ + s i n 3 θ = 1

Commented by Prithwish sen last updated on 07/Jul/19

\sin 5 θ + \sin 3 θ = 1

(5 \sin θ - {20 \sin}^{3} θ + {16 \sin}^{5} θ) + (3 \sin θ - {4 \sin}^{3} θ) = 1

\Rightarrow {16 \sin}^{5} θ - {24 \sin}^{3} θ + 8 \sin θ - 1 = 0

It is an equation of 5^{th} degree . I {don}^{'} t think

it is an easy one to solve .

Commented by Rio Michael last updated on 07/Jul/19

r e a l l y i t h i n k i g o t a d i f f i c u l t y i n i t

Answered by MJS last updated on 07/Jul/19

no exact solution possible

for x \in [0, 2 π [I get

x_{1} \approx . 132171549 (\approx 7.57287194 °)

x_{2} \approx . 620008878 (\approx 35.5238919 °)

x_{3} = π - x_{2} \approx 2.52158378 (\approx 144.476108 °)

x_{4} = π - x_{1} \approx 3.00942110 (\approx 172.427128 °)

\Rightarrow generally \underset{i = 1}{\overset{4}{x_{i}}} + 2 n π \land n \in Z (= \underset{i = 1}{\overset{4}{x_{i}}} ° + n \times 360 ° \land n \in Z)

Commented by Prithwish sen last updated on 08/Jul/19

Thank you sir