What is the general solution for 4cos(x)tan(x)-3tan(x)=0 ?

The general solution for 4cos(x)tan(x)-3tan(x)=0 is x=pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin

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4cos(x)tan(x)-3tan(x)=0

4 cos (x) tan (x) - 3 tan (x) = 0

x = πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 41.40962 \dots^{\circ} + 36 0^{\circ} n, x = 318.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 cos (x) tan (x) - 3 tan (x) = 0

Factor

4 cos (x) tan (x) - 3 tan (x) : tan (x) (4 cos (x) - 3)

4 cos (x) tan (x) - 3 tan (x)

Factor out common term

tan (x)

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

tan (x) (4 cos (x) - 3) = 0

Solving each part separately

tan (x) = 0 or 4 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

4 cos (x) - 3 = 0 : x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

4 cos (x) - 3 = 0

Move

3

to the right side

4 cos (x) - 3 = 0

Add

3

to both sides

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

Simplify

4 cos (x) = 3

4 cos (x) = 3

Divide both sides by

4

4 cos (x) = 3

Divide both sides by

4

\frac{4 cos ( x )}{4} = \frac{3}{4}

Simplify

cos (x) = \frac{3}{4}

cos (x) = \frac{3}{4}

Apply trig inverse properties

cos (x) = \frac{3}{4}

General solutions for

cos (x) = \frac{3}{4}

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

x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Combine all the solutions

x = πn, x = arccos (\frac{3}{4}) + 2 πn, x = 2 π - arccos (\frac{3}{4}) + 2 πn

Show solutions in decimal form

x = πn, x = 0.72273 \dots + 2 πn, x = 2 π - 0.72273 \dots + 2 πn

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

sin (x) = 2 cos (x)

sec (x) tan (x) = 0

2 cos (x) + 1 = 3

sec (x) + tan (x) = 0

What is the general solution for 4cos(x)tan(x)-3tan(x)=0 ?
The general solution for 4cos(x)tan(x)-3tan(x)=0 is x=pin,x=0.72273…+2pin,x=2pi-0.72273…+2pin