What is the general solution for 5tan(x)=9sin(x) ?

The general solution for 5tan(x)=9sin(x) is x=2pin,x=pi+2pin,x=0.98176…+2pin,x=2pi-0.98176…+2pin

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5tan(x)=9sin(x)

5 tan (x) = 9 sin (x)

x = 2 πn, x = π + 2 πn, x = 0.98176 \dots + 2 πn, x = 2 π - 0.98176 \dots + 2 πn

Solution steps

5 tan (x) = 9 sin (x)

Subtract

9 sin (x)

from both sides

5 tan (x) - 9 sin (x) = 0

Express with sin, cos

\frac{5 sin ( x ) - 9 cos ( x ) sin ( x )}{cos ( x )} = 0

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

5 sin (x) - 9 cos (x) sin (x) = 0

Factor

5 sin (x) - 9 cos (x) sin (x) : - sin (x) (9 cos (x) - 5)

- sin (x) (9 cos (x) - 5) = 0

Solving each part separately

sin (x) = 0 or 9 cos (x) - 5 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

9 cos (x) - 5 = 0 : x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{9}) + 2 πn, x = 2 π - arccos (\frac{5}{9}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.98176 \dots + 2 πn, x = 2 π - 0.98176 \dots + 2 πn

x, sin (\frac{π x}{4}) = y - 1

sin (θ) = \frac{12}{37}

8 tan (\frac{θ}{2}) + 8 cos (θ) tan (\frac{θ}{2}) = 1

m (0) = sin (2 x) + 2

6 cos^{2} (x) + cos (x) - 1 = 0

What is the general solution for 5tan(x)=9sin(x) ?
The general solution for 5tan(x)=9sin(x) is x=2pin,x=pi+2pin,x=0.98176…+2pin,x=2pi-0.98176…+2pin