Question

prove that, sin 18 ° = square root5-1/4; cos 36° = square root 5 +1/4.
please answer this question step by step.​

Answer 1

Answer:

Step-by-step 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