cos(3/2 pi-x)+sin(4x)=0

Solution

cos (\frac{3}{2} π - x) + sin (4 x) = 0

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Solution steps

cos (\frac{3}{2} π - x) + sin (4 x) = 0

Rewrite using trig identities

2 cos (\frac{5 x}{2}) sin (\frac{3 x}{2}) = 0

Solving each part separately

cos (\frac{5 x}{2}) = 0 or sin (\frac{3 x}{2}) = 0

cos (\frac{5 x}{2}) = 0 : x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Combine all the solutions

x = \frac{π}{5} + \frac{4 πn}{5}, x = \frac{3 π}{5} + \frac{4 πn}{5}, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

tan (x) = \frac{30}{100}

cos (θ) = - \frac{53}{8}

\frac{sin ( 15 0 ^{\circ} )}{500} = \frac{sin ( x )}{45}

sec (x) - tan (x) = 2

sin (\frac{2 π ( t + 101.75 )}{365}) = - 1