MATHEMATICS MIDDLE SCHOOL

Prove that sin(Π÷14)sin(3Π÷14)sin(5Π÷14)=2​

Answers

Answer 1
Answer:

Answer:

The correct prove will be:

sin((\pi )/(14))sin((3\pi )/(14))sin((5\pi )/(14))= (1)/(8)

Step-by-step explanation:

    sin((\pi )/(14))sin((3\pi )/(14))sin((5\pi )/(14))

            Multiply and Divide by 2cos((\pi )/(14))

= (2sin((\pi )/(14))cos((\pi )/(14))sin((3\pi )/(14))sin((5\pi )/(14)))/(2cos((\pi )/(14)))

⇒ Let (\pi )/(14) = Ф  and 7Ф = (\pi )/(2)

= sin2Ф sin3Ф sin5Ф ÷ 2cosФ

= 1/2(2sin2Фsin5Ф)sin3Ф ÷ 2cosФ

= (cos3Ф - cos7Ф) sin3Ф ÷ 4cosФ

= 1/2(2cos3Ф sin3Ф) ÷ 4cosФ               ∵cos7Ф = 0      

= sin6Ф ÷ 8cosФ

= sin(7Ф - Ф) ÷ 8cosФ                              

= (sin7ФcosФ - cos7ФsinФ) ÷ 8cosФ      ∵sin7Ф = 1  

= cosФ ÷ 8cosФ

= 1/8

Hence, The correct prove will be:

sin((\pi )/(14))sin((3\pi )/(14))sin((5\pi )/(14))= (1)/(8)

Keywords: prove, sinФ, cosФ

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What is Division?

A division is a process of splitting a specific amount into equal parts.

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