Question

If A=sin 2π÷7 + sin 4π÷7 +sin 8π÷7
   B=cos 2π÷7 + cos 4π÷7 +cos 8π÷7

Find √(A²+B²)

Answer 1

Formula :
1. $(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx$
2. $sin^{2}( \alpha ) + cos^{2}( \alpha ) = 1$
3. $sin( \alpha )sin( \beta ) + cos( \alpha )cos( \beta ) = cos( \alpha - \beta )$


$A = sin( \frac{2 \pi }{7} ) + sin( \frac{4 \pi }{7} ) + sin( \frac{8 \pi }{7} ) \\ \\ A^{2} = sin^{2}( \frac{2 \pi }{7} ) + sin^{2}( \frac{4 \pi }{7} ) + sin^{2}( \frac{8 \pi }{7} ) + 2sin( \frac{2 \pi }{7} )sin( \frac{4 \pi }{7} ) + 2sin( \frac{4 \pi }{7} )sin( \frac{8 \pi }{7} ) + 2sin( \frac{8 \pi }{7} )sin( \frac{2 \pi }{7} ) \\ \\B = cos( \frac{2 \pi }{7} ) + cos( \frac{4 \pi }{7} ) + cos( \frac{8 \pi }{7} )$

$B^{2} = cos^{2}( \frac{2 \pi }{7} ) + cos^{2}( \frac{4 \pi }{7} ) + cos^{2}( \frac{8 \pi }{7} ) + 2cos( \frac{2 \pi }{7} )cos( \frac{4 \pi }{7} ) + 2cos( \frac{4 \pi }{7} )cos( \frac{8 \pi }{7} ) + 2cos( \frac{8 \pi }{7} )cos( \frac{2 \pi }{7} )$

So,
$A^{2} + B^{2} = (sin^{2}( \frac{2 \pi }{7} ) + cos^{2}( \frac{2 \pi }{7} )) + (sin^{2}( \frac{4 \pi }{7} ) + cos^{2}( \frac{4 \pi }{7} )) + (sin^{2}( \frac{8 \pi }{7} ) + cos^{2}( \frac{8 \pi }{7} )) + 2(sin( \frac{2 \pi }{7} )sin( \frac{4 \pi }{7} ) + cos( \frac{2 \pi }{7} )cos( \frac{4 \pi }{7} )) + 2(sin( \frac{4 \pi }{7} )sin( \frac{8 \pi }{7} ) + cos( \frac{4 \pi }{7} )cos( \frac{8 \pi }{7} )) + 2(sin( \frac{8 \pi }{7} )sin( \frac{2 \pi }{7} ) + cos( \frac{8 \pi }{7} )cos( \frac{2 \pi }{7} ))$

$A^{2} + B^{2} = 1 + 1 + 1 + 2cos( \frac{2 \pi }{7} ) + 2cos( \frac{4 \pi }{7} ) + 2cos( \frac{6 \pi }{7} ) \\ \\ A^{2} + B^{2} = 3 + 2(cos( \frac{2 \pi }{7} ) + cos( \frac{6 \pi }{7} )) + 2cos( \frac{4 \pi }{7} ) \\ \\A^{2} + B^{2} = 3 + 2(2cos( \frac{4 \pi }{7} )cos( \frac{2 \pi }{7} )) + 2cos( \frac{4 \pi }{7} )$

$A^{2} + B^{2} = 3 + 4cos( \frac{4 \pi }{7})cos( \frac{2 \pi }{7} ) + 2cos( \frac{4 \pi }{7} ) \\ \\A^{2} + B^{2} = 3 + 2cos( \frac{4 \pi }{7})(2cos( \frac{2 \pi }{7} ) + 1)\\ \\\sqrt{A^{2} + B^{2}} =\sqrt{3 + 2cos( \frac{4 \pi }{7})(2cos( \frac{2 \pi }{7} ) + 1)}$

Answer 2

Answer:

Expand the expression -4( 8π + 7)

Step-by-step explanation: