What is the general solution for 5cos(x)tan(x)+3tan(x)=0 ?

The general solution for 5cos(x)tan(x)+3tan(x)=0 is x=pin,x=2.21429…+2pin,x=-2.21429…+2pin

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5cos(x)tan(x)+3tan(x)=0

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

x = πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 126.86989 \dots^{\circ} + 36 0^{\circ} n, x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Factor

5 cos (x) tan (x) + 3 tan (x) : tan (x) (5 cos (x) + 3)

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

Factor out common term

tan (x)

= tan (x) (5 cos (x) + 3)

tan (x) (5 cos (x) + 3) = 0

Solving each part separately

tan (x) = 0 or 5 cos (x) + 3 = 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

5 cos (x) + 3 = 0 : x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

5 cos (x) + 3 = 0

Move

3

to the right side

5 cos (x) + 3 = 0

Subtract

3

from both sides

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

Simplify

5 cos (x) = - 3

5 cos (x) = - 3

Divide both sides by

5

5 cos (x) = - 3

Divide both sides by

5

\frac{5 cos ( x )}{5} = \frac{- 3}{5}

Simplify

cos (x) = - \frac{3}{5}

cos (x) = - \frac{3}{5}

Apply trig inverse properties

cos (x) = - \frac{3}{5}

General solutions for

cos (x) = - \frac{3}{5}

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

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Combine all the solutions

x = πn, x = arccos (- \frac{3}{5}) + 2 πn, x = - arccos (- \frac{3}{5}) + 2 πn

Show solutions in decimal form

x = πn, x = 2.21429 \dots + 2 πn, x = - 2.21429 \dots + 2 πn

2 sec (θ) + 2 = 6

4 cos (x) tan (x) = 3 tan (x)

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

2 sin (x - \frac{π}{6}) + 1 = 0

3 - 3 sin (θ) = 6 cos^{2} (θ)

What is the general solution for 5cos(x)tan(x)+3tan(x)=0 ?
The general solution for 5cos(x)tan(x)+3tan(x)=0 is x=pin,x=2.21429…+2pin,x=-2.21429…+2pin