What is the general solution for tan(x)*sin(x)=5219 ?

The general solution for tan(x)*sin(x)=5219 is x=1.57060…+2pin,x=2pi-1.57060…+2pin

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tan(x)*sin(x)=5219

tan (x) \cdot sin (x) = 5219

x = 1.57060 \dots + 2 πn, x = 2 π - 1.57060 \dots + 2 πn

Solution steps

tan (x) sin (x) = 5219

Subtract

5219

from both sides

tan (x) sin (x) - 5219 = 0

Express with sin, cos

\frac{sin ( x )}{cos ( x )} sin (x) - 5219 = 0

Simplify

\frac{sin ( x )}{cos ( x )} sin (x) - 5219 : \frac{sin ^{2} ( x ) - 5219 cos ( x )}{cos ( x )}

\frac{sin ^{2} ( x ) - 5219 cos ( x )}{cos ( x )} = 0

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

sin^{2} (x) - 5219 cos (x) = 0

Add

5219 cos (x)

to both sides

sin^{2} (x) = 5219 cos (x)

Square both sides

(sin^{2} (x))^{2} = (5219 cos (x))^{2}

Subtract

(5219 cos (x))^{2}

from both sides

sin^{4} (x) - 27237961 cos^{2} (x) = 0

Factor

sin^{4} (x) - 27237961 cos^{2} (x) : (sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x))

(sin^{2} (x) + 5219 cos (x)) (sin^{2} (x) - 5219 cos (x)) = 0

Solving each part separately

sin^{2} (x) + 5219 cos (x) = 0 or sin^{2} (x) - 5219 cos (x) = 0

sin^{2} (x) + 5219 cos (x) = 0 : x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn

sin^{2} (x) - 5219 cos (x) = 0 : x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Combine all the solutions

x = arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = - arccos (- \frac{- 5219 + 27237965}{2}) + 2 πn, x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Verify solutions by plugging them into the original equation

x = arccos (\frac{27237965 - 5219}{2}) + 2 πn, x = 2 π - arccos (\frac{27237965 - 5219}{2}) + 2 πn

Show solutions in decimal form

x = 1.57060 \dots + 2 πn, x = 2 π - 1.57060 \dots + 2 πn

sin (z) = 2

2 \cdot sin^{2} (x) + sin (x) = 1

sin (x) = \frac{9}{10}

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

sin^{2} (x) = \frac{1 - cos ^{2} ( x )}{2}

What is the general solution for tan(x)*sin(x)=5219 ?
The general solution for tan(x)*sin(x)=5219 is x=1.57060…+2pin,x=2pi-1.57060…+2pin