What is the general solution for tan(x)=2cos(x)tan(x) ?

The general solution for tan(x)=2cos(x)tan(x) is x=pin,x= pi/3+2pin,x=(5pi)/3+2pin

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Popular >

tan(x)=2cos(x)tan(x)

Solution

tan (x) = 2 cos (x) tan (x)

Solution

x = πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

tan (x) = 2 cos (x) tan (x)

Subtract

2 cos (x) tan (x)

from both sides

tan (x) - 2 cos (x) tan (x) = 0

Factor

tan (x) - 2 cos (x) tan (x) : - tan (x) (2 cos (x) - 1)

tan (x) - 2 cos (x) tan (x)

Factor out common term

- tan (x)

= - tan (x) (- 1 + 2 cos (x))

- tan (x) (2 cos (x) - 1) = 0

Solving each part separately

tan (x) = 0 or 2 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

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 cos (x) - 1 = 0

Move

1

to the right side

2 cos (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 cos (x) = 1

2 cos (x) = 1

Divide both sides by

2

2 cos (x) = 1

Divide both sides by

2

\frac{2 cos ( x )}{2} = \frac{1}{2}

Simplify

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \frac{1}{2}

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}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Popular Examples

tan(θ)=2.6

tan (θ) = 2.6

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

2 sin^{2} (x) = 1 + cos (x)

sin(θ)= 8/17

sin (θ) = \frac{8}{17}

cos^2(x)-3sin(x)=3

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

7sin(x)cos(x)=8cos(x)

7 sin (x) cos (x) = 8 cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for tan(x)=2cos(x)tan(x) ?
The general solution for tan(x)=2cos(x)tan(x) is x=pin,x= pi/3+2pin,x=(5pi)/3+2pin