What is the general solution for 10sin(x)tan(x)-9tan(x)=0 ?

The general solution for 10sin(x)tan(x)-9tan(x)=0 is x=pin,x=1.11976…+2pin,x=pi-1.11976…+2pin

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10sin(x)tan(x)-9tan(x)=0

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

x = πn, x = 1.11976 \dots + 2 πn, x = π - 1.11976 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 64.15806 \dots^{\circ} + 36 0^{\circ} n, x = 115.84193 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

10 sin (x) tan (x) - 9 tan (x) : tan (x) (10 sin (x) - 9)

10 sin (x) tan (x) - 9 tan (x)

Factor out common term

tan (x)

= tan (x) (10 sin (x) - 9)

tan (x) (10 sin (x) - 9) = 0

Solving each part separately

tan (x) = 0 or 10 sin (x) - 9 = 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

10 sin (x) - 9 = 0 : x = arcsin (\frac{9}{10}) + 2 πn, x = π - arcsin (\frac{9}{10}) + 2 πn

10 sin (x) - 9 = 0

Move

9

to the right side

10 sin (x) - 9 = 0

Add

9

to both sides

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

Simplify

10 sin (x) = 9

10 sin (x) = 9

Divide both sides by

10

10 sin (x) = 9

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{9}{10}

Simplify

sin (x) = \frac{9}{10}

sin (x) = \frac{9}{10}

Apply trig inverse properties

sin (x) = \frac{9}{10}

General solutions for

sin (x) = \frac{9}{10}

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

x = arcsin (\frac{9}{10}) + 2 πn, x = π - arcsin (\frac{9}{10}) + 2 πn

x = arcsin (\frac{9}{10}) + 2 πn, x = π - arcsin (\frac{9}{10}) + 2 πn

Combine all the solutions

x = πn, x = arcsin (\frac{9}{10}) + 2 πn, x = π - arcsin (\frac{9}{10}) + 2 πn

Show solutions in decimal form

x = πn, x = 1.11976 \dots + 2 πn, x = π - 1.11976 \dots + 2 πn

- sin (t) = 0

cos (x) = 4

tan (x) = \frac{5}{6}

2 cos (θ) = - 3

cos (t) = \frac{3}{2}

What is the general solution for 10sin(x)tan(x)-9tan(x)=0 ?
The general solution for 10sin(x)tan(x)-9tan(x)=0 is x=pin,x=1.11976…+2pin,x=pi-1.11976…+2pin