What is the general solution for 4sin(x)=3tan(x) ?

The general solution for 4sin(x)=3tan(x) is x=2pin,x=pi+2pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin

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4sin(x)=3tan(x)

4 sin (x) = 3 tan (x)

x = 2 πn, x = π + 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

Solution steps

4 sin (x) = 3 tan (x)

Subtract

3 tan (x)

from both sides

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

Express with sin, cos

\frac{- 3 sin ( x ) + 4 cos ( x ) sin ( x )}{cos ( x )} = 0

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

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

Factor

- 3 sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) - 3)

sin (x) (4 cos (x) - 3) = 0

Solving each part separately

sin (x) = 0 or 4 cos (x) - 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

4 cos (x) - 3 = 0 : x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

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

x, c sin (m x) m = \frac{nπ}{a}

x, - 0.3 \cdot cos (x) + sin (x) = 0.10618

6 tan (x) = 4

3 tan (x) + cot (x) - 5 csc (x) = 0

What is the general solution for 4sin(x)=3tan(x) ?
The general solution for 4sin(x)=3tan(x) is x=2pin,x=pi+2pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin