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

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

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

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

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

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 109.47122 \dots^{\circ} + 36 0^{\circ} n, x = - 109.47122 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

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

3 cos (x) tan (x) + tan (x)

Factor out common term

tan (x)

= tan (x) (3 cos (x) + 1)

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

Solving each part separately

tan (x) = 0 or 3 cos (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

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

3 cos (x) + 1 = 0

Move

1

to the right side

3 cos (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

3 cos (x) = - 1

3 cos (x) = - 1

Divide both sides by

3

3 cos (x) = - 1

Divide both sides by

3

\frac{3 cos ( x )}{3} = \frac{- 1}{3}

Simplify

cos (x) = - \frac{1}{3}

cos (x) = - \frac{1}{3}

Apply trig inverse properties

cos (x) = - \frac{1}{3}

General solutions for

cos (x) = - \frac{1}{3}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

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

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

Combine all the solutions

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

Show solutions in decimal form

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

sin (x) = \frac{3}{2}

3 sin (x) = 1 + cos (2 x)

2 cos^{2} (θ) + 3 cos (θ) = 0

5 (sin (θ) + 1) = 5

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

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