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

The general solution for 10sin(x)tan(x)+tan(x)=0 is x=pin,x=-0.10016…+2pin,x=pi+0.10016…+2pin

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10sin(x)tan(x)+tan(x)=0

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

x = πn, x = - 0.10016 \dots + 2 πn, x = π + 0.10016 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = - 5.73917 \dots^{\circ} + 36 0^{\circ} n, x = 185.73917 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

10 sin (x) tan (x) + tan (x) : tan (x) (10 sin (x) + 1)

10 sin (x) tan (x) + tan (x)

Factor out common term

tan (x)

= tan (x) (10 sin (x) + 1)

tan (x) (10 sin (x) + 1) = 0

Solving each part separately

tan (x) = 0 or 10 sin (x) + 1 = 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) + 1 = 0 : x = arcsin (- \frac{1}{10}) + 2 πn, x = π + arcsin (\frac{1}{10}) + 2 πn

10 sin (x) + 1 = 0

Move

1

to the right side

10 sin (x) + 1 = 0

Subtract

1

from both sides

10 sin (x) + 1 - 1 = 0 - 1

Simplify

10 sin (x) = - 1

10 sin (x) = - 1

Divide both sides by

10

10 sin (x) = - 1

Divide both sides by

10

\frac{10 sin ( x )}{10} = \frac{- 1}{10}

Simplify

sin (x) = - \frac{1}{10}

sin (x) = - \frac{1}{10}

Apply trig inverse properties

sin (x) = - \frac{1}{10}

General solutions for

sin (x) = - \frac{1}{10}

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

x = arcsin (- \frac{1}{10}) + 2 πn, x = π + arcsin (\frac{1}{10}) + 2 πn

x = arcsin (- \frac{1}{10}) + 2 πn, x = π + arcsin (\frac{1}{10}) + 2 πn

Combine all the solutions

x = πn, x = arcsin (- \frac{1}{10}) + 2 πn, x = π + arcsin (\frac{1}{10}) + 2 πn

Show solutions in decimal form

x = πn, x = - 0.10016 \dots + 2 πn, x = π + 0.10016 \dots + 2 πn

2 sin^{2} (θ) + sin (θ) - 1 = 0, 0 \leq θ \leq 2 π

10 sin (x) cos (x) - 7 cos (x) = 0

sin (x + \frac{π}{4}) - sin (x - \frac{π}{4}) = 1

2 cos^{3} (x) = 2 cos (x)

sin^{2} (x) + cos^{2} (x) = 2

What is the general solution for 10sin(x)tan(x)+tan(x)=0 ?
The general solution for 10sin(x)tan(x)+tan(x)=0 is x=pin,x=-0.10016…+2pin,x=pi+0.10016…+2pin