Question

prove that :

sin 18 = (root 5)-1 / 4

Answer 1

sin 54° = cos 36° since they're complementary angles 
If x = 18°, and 3·18° = 54°and 2·18° = 36° then 
sin(3x) = cos(2x). ← That's the magic step, leading to... 
sin(3x) = sin(2x + x) = sin(2x)cos(x) + sinx cos(2x) 
thus, 
sin(3x) = cos(2x) 
sin(2x)cos(x) + sinx cos(2x) = cos(2x) 
sin(2x)cos(x) = cos(2x) - sinx cos(2x) 
sin(2x)cos(x) = cos(2x) (1-sinx) 
2sinxcos²x = (1-2sin²x)(1-sinx) 
2sinx(1-sin²x) = (1-2sin²x)(1-sinx) 
2sinx(1-sinx)(1+sinx) = (1-2sin²x)(1-sinx) 
Since we know sinx≠1, we can divide by 1-sinx 
2sinx(1+sinx) = (1-2sin²x) 
2sinx + 2sin²x = 1 - 2sin²x 
4sin²x + 2sinx - 1 = 0 
sinx = (-1±√5)/4 
so sin18° = (-1+√5)/4 or (-1-√5)/4 
but we know sin 18° is positive, leaving us with sin18° =(-1+√5)/4

Answer 2

Answer:

Explanation:sin 54° = cos 36° since they're complementary angles 

If x = 18°, and 3·18° = 54°and 2·18° = 36° then 

sin(3x) = cos(2x). ← That's the magic step, leading to... 

sin(3x) = sin(2x + x) = sin(2x)cos(x) + sinx cos(2x) 

thus, 

sin(3x) = cos(2x) 

sin(2x)cos(x) + sinx cos(2x) = cos(2x) 

sin(2x)cos(x) = cos(2x) - sinx cos(2x) 

sin(2x)cos(x) = cos(2x) (1-sinx) 

2sinxcos²x = (1-2sin²x)(1-sinx) 

2sinx(1-sin²x) = (1-2sin²x)(1-sinx) 

2sinx(1-sinx)(1+sinx) = (1-2sin²x)(1-sinx) 

Since we know sinx≠1, we can divide by 1-sinx 

2sinx(1+sinx) = (1-2sin²x) 

2sinx + 2sin²x = 1 - 2sin²x 

4sin²x + 2sinx - 1 = 0 

sinx = (-1±√5)/4 

so sin18° = (-1+√5)/4 or (-1-√5)/4 

but we know sin 18° is positive, leaving us with sin18° =(-1+√5)/4

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