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Sin 2π/7+sin4π/7+sin8π/7 . it`s numeric value

Sin 2π/7+sin4π/7+sin8π/7 . it`s numeric value

Grade:11

3 Answers

Nishant pandey
39 Points
5 years ago
We seek S = sin (2π/7) + sin (4π/7) - sin (π/7) 
Let s = sin(π/7), c = cos(π/7) so s² + c² = 1. 

From cos(α + β) and sin(α + β) formulas come 
cos(2π/7) = c² - s² = 2c² - 1 
sin(2π/7) = 2cs 
sin(3π/7) = s(2c² - 1) + c(2cs) = s(4c² - 1), and 
sin(4π/7) = 4cs(2c²-1) 

Since sin(3π/7) = sin(4π/7), we have 4c² - 1 = 4c(2c² - 1) or 
(1): 8c³ = 4c² + 4c - 1 
This means any polynomial in c can be reduced to a quadratic or less. 

S² = (sin(2π/7) + sin(4π/7) - sin(π/7))² 
= (sin(2π/7) + sin(3π/7) - sin(π/7))² 
= (2sc + s(4c²-1) - s)² 
= s² * (4c² + 2c -2)² 
= (1-c²) * (4c² + 2c -2)² which was the reason for squaring S: a polynomial in c, 
= (4c² + 2c - 2)² - (4c³ + 2c² - 2c)² which we now reduce. First, apply (1) to 4c³ to get 
= (4c² + 2c - 2)² - (4c² - ½)² which, by difference of squares is 
= (8c² + 2c - 5/2) * (2c - 3/2) 
= 16c³ - 8c² - 8c + 15/4 and now apply (1) a final time 
= -2 + 15/4 
= 7/4 

Hence, the value of the original expression is S = ½ * √7
Raghu Vamshi Hemadri
72 Points
5 years ago
We seek S = sin (2π/7) + sin (4π/7) - sin (π/7) 
Let s = sin(π/7), c = cos(π/7) so s² + c² = 1. 

From cos(α + β) and sin(α + β) formulas come 
cos(2π/7) = c² - s² = 2c² - 1 
sin(2π/7) = 2cs 
sin(3π/7) = s(2c² - 1) + c(2cs) = s(4c² - 1), and 
sin(4π/7) = 4cs(2c²-1) 

Since sin(3π/7) = sin(4π/7), we have 4c² - 1 = 4c(2c² - 1) or 
(1): 8c³ = 4c² + 4c - 1 
This means any polynomial in c can be reduced to a quadratic or less. 

S² = (sin(2π/7) + sin(4π/7) - sin(π/7))² 
= (sin(2π/7) + sin(3π/7) - sin(π/7))² 
= (2sc + s(4c²-1) - s)² 
= s² * (4c² + 2c -2)² 
= (1-c²) * (4c² + 2c -2)² which was the reason for squaring S: a polynomial in c, 
= (4c² + 2c - 2)² - (4c³ + 2c² - 2c)² which we now reduce. First, apply (1) to 4c³ to get 
= (4c² + 2c - 2)² - (4c² - ½)² which, by difference of squares is 
= (8c² + 2c - 5/2) * (2c - 3/2) 
= 16c³ - 8c² - 8c + 15/4 and now apply (1) a final time 
= -2 + 15/4 
= 7/4 

Hence, the value of the original expression is S = ½ * √7
SHAIK HAFEEZUL KAREEM
109 Points
5 years ago
 
We seek S = sin (2π/7) + sin (4π/7) - sin (π/7) 
Let s = sin(π/7), c = cos(π/7) so s² + c² = 1. 

From cos(α + β) and sin(α + β) formulas come 
cos(2π/7) = c² - s² = 2c² - 1 
sin(2π/7) = 2cs 
sin(3π/7) = s(2c² - 1) + c(2cs) = s(4c² - 1), and 
sin(4π/7) = 4cs(2c²-1) 

Since sin(3π/7) = sin(4π/7), we have 4c² - 1 = 4c(2c² - 1) or 
(1): 8c³ = 4c² + 4c - 1 
This means any polynomial in c can be reduced to a quadratic or less. 

S² = (sin(2π/7) + sin(4π/7) - sin(π/7))² 
= (sin(2π/7) + sin(3π/7) - sin(π/7))² 
= (2sc + s(4c²-1) - s)² 
= s² * (4c² + 2c -2)² 
= (1-c²) * (4c² + 2c -2)² which was the reason for squaring S: a polynomial in c, 
= (4c² + 2c - 2)² - (4c³ + 2c² - 2c)² which we now reduce. First, apply (1) to 4c³ to get 
= (4c² + 2c - 2)² - (4c² - ½)² which, by difference of squares is 
= (8c² + 2c - 5/2) * (2c - 3/2) 
= 16c³ - 8c² - 8c + 15/4 and now apply (1) a final time 
= -2 + 15/4 
= 7/4 

Hence, the value of the original expression is S = ½ * √7

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