What is the general solution for 2tan(x)-3sin(x)=0 ?

The general solution for 2tan(x)-3sin(x)=0 is x=2pin,x=pi+2pin,x=0.84106…+2pin,x=2pi-0.84106…+2pin

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2tan(x)-3sin(x)=0

2 tan (x) - 3 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Solution steps

2 tan (x) - 3 sin (x) = 0

Express with sin, cos

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

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

2 sin (x) - 3 cos (x) sin (x) = 0

Factor

2 sin (x) - 3 cos (x) sin (x) : - sin (x) (3 cos (x) - 2)

- sin (x) (3 cos (x) - 2) = 0

Solving each part separately

sin (x) = 0 or 3 cos (x) - 2 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{2}{3}) + 2 πn, x = 2 π - arccos (\frac{2}{3}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

sin (x) + 5 = 4

- 9 cos (x) = 0

- 3 cos (x) = 0

4 cot (x) + 2 = 6

4 sin^{2} (θ) - 4 = - 3

What is the general solution for 2tan(x)-3sin(x)=0 ?
The general solution for 2tan(x)-3sin(x)=0 is x=2pin,x=pi+2pin,x=0.84106…+2pin,x=2pi-0.84106…+2pin