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

The general solution for 4sin(x)=tan(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin

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4sin(x)=tan(x)

4 sin (x) = tan (x)

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Solution steps

4 sin (x) = tan (x)

Subtract

tan (x)

from both sides

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

Express with sin, cos

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

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

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

Factor

- sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) - 1)

sin (x) (4 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 4 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

sin (a) = \frac{7}{25}

3 sin^{2} (x) + 2 = 5 sin (x)

sin (θ) = 0.65

2 = - 4 cos (θ)

3 \cdot sin (x) = 3 \cdot cos (x)

What is the general solution for 4sin(x)=tan(x) ?
The general solution for 4sin(x)=tan(x) is x=2pin,x=pi+2pin,x=1.31811…+2pin,x=2pi-1.31811…+2pin