Question

1. sin6x/cos8x=-1

2. sin12x/sin8x=-1

Answer 1

 

$1. frac{sin6x}{cos8x} = -1\\ cos8x ne 0; 8x ne frac{pi}{2} + pi n, n in Z; x ne frac{pi}{16} + frac{pi n}{8}, n in Z\\ sin6x = -cos8x\\ sin6x + cos8x = 0\\ cos(frac{pi}{2} - 6x) + cos8x = 0\\ 2cos(frac{frac{pi}{2} - 6x + 8x}{2})cos(frac{frac{pi}{2} - 6x - 8x }{2}) = 0\\ cos(frac{pi}{4} + x)cos(frac{pi}{4} - 7x) = 0$

 

 

$</var>1) cos(frac{pi}{4} x) = 0\\ frac{pi}{4} x = frac{pi}{2} pi n, n in Z\\ x = frac{pi}{4} pi n, n in Z \\ frac{pi}{4} pi n ne frac{pi}{16} frac{pi k}{8}\\ n ne frac{2k - 3}{16}\\ boxed{ x = frac{pi}{4} pi n, n in Z setminus <var>{<var> frac{2k - 3}{16}| k in Z </var>}</var> }$</var></p> <p> </p> <p> </p> <p><var></var><img src=[/tex]2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} ne frac{pi}{16} + frac{pi k}{8}\\ n ne -frac{14k + 11}{16}\\ boxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z setminus { -frac{14k + 11}{16}| k in Z } }" title="2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} ne frac{pi}{16} + frac{pi k}{8}\\ n ne -frac{14k + 11}{16}\\ boxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z setminus { -frac{14k + 11}{16}| k in Z } }" alt="2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} ne frac{pi}{16} + frac{pi k}{8}\\ n ne -frac{14k + 11}{16}\\ boxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z setminus { -frac{14k + 11}{16}| k in Z } }" />

 

 

 

$<var>2. frac{sin12x}{sin8x} = -1\\ sin8x ne 0; 8x ne pi n, n in Z; x ne frac{pi n}{8}, n in Z\\ sin12x + sin8x = 0\\ 2sin(frac{12x + 8x}{2})cos(frac{12x-8x}{2}) = 0\\ sin(10x)cos(4x) = 0\\\ 1) sin(10x) = 0\\ 10x = pi n, n in Z,\\ frac{pi n}{10} ne frac{pi k}{8}\\ n ne frac{5k}{4} \\ boxed{ x = frac{pi n}{10}, n in Z setminus { <var>frac{5k}{4}</var> | k in Z } }$

 

 

$<var>2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} ne frac{pi}{16} + frac{pi k}{8}\\ n ne -frac{14k + 11}{16}\\ boxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z <var>setminus <var>{ <var><var>-frac{14k + 11}{16}</var>| k in Z</var> </var></var>}</var> }$

 

 

 

$</var>1) cos(frac{pi}{4} + x) = 0\\ frac{pi}{4} + x = frac{pi}{2} + pi n, n in Z\\ x = frac{pi}{4} + pi n, n in Z \\ frac{pi}{4} + pi n ne frac{pi}{16} + frac{pi k}{8}\\ n ne frac{2k - 3}{16}\\ boxed{ x = frac{pi}{4} + pi n, n in Z setminus <var>{<var> frac{2k - 3}{16}| k in Z </var>}</var> }$

 

 

$<var>2) cos(frac{pi}{4} - 7x) = 0\\ frac{pi}{4} - 7x =frac{pi}{2} + pi n, n in Z\\ -7x = pi n + frac{pi}{4}, n in Z\\ -frac{pi n}{7} - frac{pi}{28} ne frac{pi}{16} + frac{pi k}{8}\\ n ne -frac{14k + 11}{16}\\ boxed{ x = -frac{pi n}{7} - frac{pi}{28}, n in Z <var>setminus <var>{ <var><var>-frac{14k + 11}{16}</var>| k in Z</var> </var></var>}</var> }$

 

 

 

$<var>2. frac{sin12x}{sin8x} = -1\\ sin8x ne 0; 8x ne pi n, n in Z; x ne frac{pi n}{8}, n in Z\\ sin12x + sin8x = 0\\ 2sin(frac{12x + 8x}{2})cos(frac{12x-8x}{2}) = 0\\ sin(10x)cos(4x) = 0\\\ 1) sin(10x) = 0\\ 10x = pi n, n in Z,\\ frac{pi n}{10} ne frac{pi k}{8}\\ n ne frac{5k}{4} \\ boxed{ x = frac{pi n}{10}, n in Z setminus { <var>frac{5k}{4}</var> | k in Z } }" /></var></p> <p> </p> <p> </p> <p>[tex]2) cos(4x) = 0\\ 4x = frac{pi}{2} + pi n, n in Z\\ x = frac{pi}{8} + frac{pi n}{4}, n in Z\\ frac{pi}{8} + frac{pi n}{4} ne <var> frac{pi k}{8}</var> \\ n ne <var>frac{k - 1}{2}</var>\\ boxed{ x = frac{pi}{8} + frac{pi n}{4}, n in Z setminus <var>{<var><var>frac{k - 1}{2}</var></var> | k in Z } }</var>$