Question

sin²π/6:sin²π/4:sin²π/3:sin²π/2=1:2:3:4
​

Answer 1

Answer:

the proof is step by step explained below

Step-by-step explanation:

Taking left hand side(LHS) only and solving,

$\sin^{2} \frac{\pi}{6} : \sin^{2} \frac{\pi}{4} : \sin^{2} \frac{\pi}{3} : \sin^{2} \frac{\pi}{2}$

Now as we know that

$\sin( {30}^{o} ) = \frac{1}{2} \\ {30}^{o} \: in \: radians \: is \: \frac{\pi}{6} \\ \sin( \frac{\pi}{6} ) = \frac{1}{2}$

To convert degree into radians this formula is used

$\: degree \times \frac{\pi}{ {180}^{o} }$

$\sin( {45}^{o} ) = \frac{1}{ \sqrt{2} } \\ {45}^{o} \: in \: radians \: is \: \frac{\pi}{4} \\ \sin( \frac{\pi}{4} ) = \frac{1}{ \sqrt{2} }$

$\sin( {60}^{o} ) = \frac{ \sqrt{3} }{2} \\ {60}^{o} \: in \: radians \: is \: \frac{\pi}{3} \\ \sin( \frac{\pi}{3} ) = \frac{ \sqrt{3} }{2}$

$\sin( {90}^{o} ) = 1 \\ {90}^{o} \: in \: radians \: is \: \frac{\pi}{2} \\ \sin( \frac{\pi}{2} ) = 1$

Now applying all the above values in Left hand side equation,

$\sin^{2} \frac{\pi}{6} : \sin^{2} \frac{\pi}{4} : \sin^{2} \frac{\pi}{3} : \sin^{2} \frac{\pi}{2} \\ = ({ \frac{1}{2} })^{2} :({ \frac{1}{ \sqrt{2} } })^{2} : ({ \frac{ \sqrt{3} }{2} })^{2} : { 1}^{2} \\ = \frac{1}{4} : \frac{1}{2} : \frac{3}{4} :1 \\ multiplying \: by \: 4 \\ = \frac{1}{4} \times 4: \frac{1}{2} \times 4: \frac{3}{4} \times 4:1 \times 4 \\ = 1:2:3:4$

Hence,

$LHS = RHS \: proved \\ 1:2:3:4 = 1:2:3:4$