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

The general solution for 3sin(x)+tan(x)=0 is x=2pin,x=pi+2pin,x=1.91063…+2pin,x=-1.91063…+2pin

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3sin(x)+tan(x)=0

3 sin (x) + tan (x) = 0

x = 2 πn, x = π + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

Solution steps

3 sin (x) + tan (x) = 0

Express with sin, cos

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

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

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

Factor

sin (x) + 3 cos (x) sin (x) : sin (x) (3 cos (x) + 1)

sin (x) (3 cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or 3 cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

3 cos (x) + 1 = 0 : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.91063 \dots + 2 πn, x = - 1.91063 \dots + 2 πn

2 sin^{4} (x) - cos^{2} (x) = 0

sin (b) = 0.4809

cos^{2} (x) - \frac{3}{4} = 0, 0 \leq x \leq 2 π

sin^{2} (x) - sin (x) - cos^{2} (x) = 0

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

What is the general solution for 3sin(x)+tan(x)=0 ?
The general solution for 3sin(x)+tan(x)=0 is x=2pin,x=pi+2pin,x=1.91063…+2pin,x=-1.91063…+2pin