What is the general solution for (tan(x)-sqrt(3))(2sin(x)-1)=0 ?

The general solution for (tan(x)-sqrt(3))(2sin(x)-1)=0 is x= pi/3+pin,x= pi/6+2pin,x=(5pi)/6+2pin

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(tan(x)-sqrt(3))(2sin(x)-1)=0

Solution

(tan (x) - 3) (2 sin (x) - 1) = 0

Solution

x = \frac{π}{3} + πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

(tan (x) - 3) (2 sin (x) - 1) = 0

Solving each part separately

tan (x) - 3 = 0 or 2 sin (x) - 1 = 0

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

tan (x) - 3 = 0

Move

3

to the right side

tan (x) - 3 = 0

Add

3

to both sides

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

Simplify

tan (x) = 3

tan (x) = 3

General solutions for

tan (x) = 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}{3} + πn

x = \frac{π}{3} + πn

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Move

1

to the right side

2 sin (x) - 1 = 0

Add

1

to both sides

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

Simplify

2 sin (x) = 1

2 sin (x) = 1

Divide both sides by

2

2 sin (x) = 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{3} + πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Popular Examples

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2sin^2(x)-5sin(x)-4=0

2 sin^{2} (x) - 5 sin (x) - 4 = 0

sin(θ)=0.4815

sin (θ) = 0.4815

6cos^2(2x)=sin(2x)+5

6 cos^{2} (2 x) = sin (2 x) + 5

sin(A)= 4/5 ,sin(2A)

sin (A) = \frac{4}{5}, sin (2 A)

Frequently Asked Questions (FAQ)

What is the general solution for (tan(x)-sqrt(3))(2sin(x)-1)=0 ?
The general solution for (tan(x)-sqrt(3))(2sin(x)-1)=0 is x= pi/3+pin,x= pi/6+2pin,x=(5pi)/6+2pin