What is the general solution for 9sin(x)tan(x)-2tan(x)=0 ?

The general solution for 9sin(x)tan(x)-2tan(x)=0 is x=pin,x=0.22409…+2pin,x=pi-0.22409…+2pin

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9sin(x)tan(x)-2tan(x)=0

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

x = πn, x = 0.22409 \dots + 2 πn, x = π - 0.22409 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 12.83958 \dots^{\circ} + 36 0^{\circ} n, x = 167.16041 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

9 sin (x) tan (x) - 2 tan (x) : tan (x) (9 sin (x) - 2)

9 sin (x) tan (x) - 2 tan (x)

Factor out common term

tan (x)

= tan (x) (9 sin (x) - 2)

tan (x) (9 sin (x) - 2) = 0

Solving each part separately

tan (x) = 0 or 9 sin (x) - 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

9 sin (x) - 2 = 0 : x = arcsin (\frac{2}{9}) + 2 πn, x = π - arcsin (\frac{2}{9}) + 2 πn

9 sin (x) - 2 = 0

Move

2

to the right side

9 sin (x) - 2 = 0

Add

2

to both sides

9 sin (x) - 2 + 2 = 0 + 2

Simplify

9 sin (x) = 2

9 sin (x) = 2

Divide both sides by

9

9 sin (x) = 2

Divide both sides by

9

\frac{9 sin ( x )}{9} = \frac{2}{9}

Simplify

sin (x) = \frac{2}{9}

sin (x) = \frac{2}{9}

Apply trig inverse properties

sin (x) = \frac{2}{9}

General solutions for

sin (x) = \frac{2}{9}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

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

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

Combine all the solutions

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

Show solutions in decimal form

x = πn, x = 0.22409 \dots + 2 πn, x = π - 0.22409 \dots + 2 πn

3 cot (3 x) + 1 = 0

tan (4 x) = 3

cos (x) + 2 = - cos (x)

- sin^{2} (x) = 0

4 cos (2 x) = 23

What is the general solution for 9sin(x)tan(x)-2tan(x)=0 ?
The general solution for 9sin(x)tan(x)-2tan(x)=0 is x=pin,x=0.22409…+2pin,x=pi-0.22409…+2pin