What is the general solution for tan(x)=5sin(x) ?

The general solution for tan(x)=5sin(x) is x=2pin,x=pi+2pin,x=1.36943…+2pin,x=2pi-1.36943…+2pin

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tan(x)=5sin(x)

tan (x) = 5 sin (x)

x = 2 πn, x = π + 2 πn, x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn

Solution steps

tan (x) = 5 sin (x)

Subtract

5 sin (x)

from both sides

tan (x) - 5 sin (x) = 0

Express with sin, cos

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

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

sin (x) - 5 cos (x) sin (x) = 0

Factor

sin (x) - 5 cos (x) sin (x) : - sin (x) (5 cos (x) - 1)

- sin (x) (5 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 5 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

5 cos (x) - 1 = 0 : x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{5}) + 2 πn, x = 2 π - arccos (\frac{1}{5}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.36943 \dots + 2 πn, x = 2 π - 1.36943 \dots + 2 πn

cot^{2} (x) = - 2 cot (x) - 1

cos (\frac{π t}{2}) = 0

sin (θ) = - \frac{3}{7}

tan (θ) = \frac{3}{- 1}

cos (x) = \frac{8}{10}

What is the general solution for tan(x)=5sin(x) ?
The general solution for tan(x)=5sin(x) is x=2pin,x=pi+2pin,x=1.36943…+2pin,x=2pi-1.36943…+2pin