What is the general solution for 9tan(x)sin(x)-5sin(x)=0 ?

The general solution for 9tan(x)sin(x)-5sin(x)=0 is x=2pin,x=pi+2pin,x=0.50709…+pin

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9tan(x)sin(x)-5sin(x)=0

9 tan (x) sin (x) - 5 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 0.50709 \dots + πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 29.05460 \dots^{\circ} + 18 0^{\circ} n

Solution steps

9 tan (x) sin (x) - 5 sin (x) = 0

Factor

9 tan (x) sin (x) - 5 sin (x) : sin (x) (9 tan (x) - 5)

9 tan (x) sin (x) - 5 sin (x)

Factor out common term

sin (x)

= sin (x) (9 tan (x) - 5)

sin (x) (9 tan (x) - 5) = 0

Solving each part separately

sin (x) = 0 or 9 tan (x) - 5 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

9 tan (x) - 5 = 0 : x = arctan (\frac{5}{9}) + πn

9 tan (x) - 5 = 0

Move

5

to the right side

9 tan (x) - 5 = 0

Add

5

to both sides

9 tan (x) - 5 + 5 = 0 + 5

Simplify

9 tan (x) = 5

9 tan (x) = 5

Divide both sides by

9

9 tan (x) = 5

Divide both sides by

9

\frac{9 tan ( x )}{9} = \frac{5}{9}

Simplify

tan (x) = \frac{5}{9}

tan (x) = \frac{5}{9}

Apply trig inverse properties

tan (x) = \frac{5}{9}

General solutions for

tan (x) = \frac{5}{9}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{5}{9}) + πn

x = arctan (\frac{5}{9}) + πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arctan (\frac{5}{9}) + πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 0.50709 \dots + πn

x, y = arctan (x)

7 cos (x) + 7 sin (x) tan (x) = 14

cot (θ) + 5 csc (θ) = 6

3 csc^{2} (x) + 2 csc (x) = 0

7 sin (2 x) sin (x) = 7 cos (x)

What is the general solution for 9tan(x)sin(x)-5sin(x)=0 ?
The general solution for 9tan(x)sin(x)-5sin(x)=0 is x=2pin,x=pi+2pin,x=0.50709…+pin