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

The general solution for sqrt(3)cos(x)tan(x)-cos(x)=0 is x= pi/6+pin

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

Solution

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

Solution

x = \frac{π}{6} + πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Factor

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

cos (x) = 0 or 3 tan (x) - 1 = 0

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

cos (x) = 0

General solutions for

cos (x) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

3 tan (x) - 1 = 0 : x = \frac{π}{6} + πn

3 tan (x) - 1 = 0

Move

1

to the right side

3 tan (x) - 1 = 0

Add

1

to both sides

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

Simplify

3 tan (x) = 1

3 tan (x) = 1

Divide both sides by

3

3 tan (x) = 1

Divide both sides by

3

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x )}{3} = \frac{1}{3}

Simplify

\frac{3 tan ( x )}{3} : tan (x)

\frac{3 tan ( x )}{3}

Cancel the common factor:

3

= tan (x)

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

tan (x) = \frac{3}{3}

General solutions for

tan (x) = \frac{3}{3}

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 = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = \frac{π}{6} + πn

Popular Examples

4sin^2(θ)=3

4 sin^{2} (θ) = 3

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

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

sec^2(x)-sec(x)-2=0

sec^{2} (x) - sec (x) - 2 = 0

tan^2(x)=0

tan^{2} (x) = 0

tan(θ)=-1/(sqrt(3))

tan (θ) = - \frac{1}{3}

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)cos(x)tan(x)-cos(x)=0 ?
The general solution for sqrt(3)cos(x)tan(x)-cos(x)=0 is x= pi/6+pin