What is the general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 ?

The general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 is x= pi/4+pin,x=1.32581…+pin

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(sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} = 0

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

Solution steps

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} = 0

Simplify

\frac{sin ( x )}{cos ( x )} - 5 + \frac{4 cos ( x )}{sin ( x )} : \frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )}

\frac{sin ^{2} ( x ) - 5 cos ( x ) sin ( x ) + 4 cos ^{2} ( x )}{cos ( x ) sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) = 0

Factor

sin^{2} (x) - 5 cos (x) sin (x) + 4 cos^{2} (x) : (sin (x) - cos (x)) (sin (x) - 4 cos (x))

(sin (x) - cos (x)) (sin (x) - 4 cos (x)) = 0

Solving each part separately

sin (x) - cos (x) = 0 or sin (x) - 4 cos (x) = 0

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

sin (x) - 4 cos (x) = 0 : x = arctan (4) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (4) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = 1.32581 \dots + πn

x, y = sin (3.2 x)

sin (x) = 0.37, π < x < \frac{π}{2}

5 cot (x) + 3 tan (x) = 8

sin (2 x) - 1 = cos (2 x), \forall 0 \leq θ < 2 π

tan (θ) = \frac{14}{20}

What is the general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 ?
The general solution for (sin(x))/(cos(x))-5+(4cos(x))/(sin(x))=0 is x= pi/4+pin,x=1.32581…+pin