What is the general solution for csc(x)tan(x)=(2sqrt(3))/3 ?

The general solution for csc(x)tan(x)=(2sqrt(3))/3 is x= pi/6+2pin,x=(11pi)/6+2pin

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csc(x)tan(x)=(2sqrt(3))/3

Solution

csc (x) tan (x) = \frac{2 3}{3}

Solution

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

csc (x) tan (x) = \frac{2 3}{3}

Subtract

\frac{2 3}{3}

from both sides

csc (x) tan (x) - \frac{2}{3} = 0

Simplify

csc (x) tan (x) - \frac{2}{3} : \frac{3 csc ( x ) tan ( x ) - 2}{3}

csc (x) tan (x) - \frac{2}{3}

Convert element to fraction:

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

= \frac{csc ( x ) tan ( x ) 3}{3} - \frac{2}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{csc ( x ) tan ( x ) 3 - 2}{3}

\frac{3 csc ( x ) tan ( x ) - 2}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 csc (x) tan (x) - 2 = 0

Express with sin, cos

- 2 + csc (x) 3 tan (x)

Use the basic trigonometric identity:

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

= - 2 + \frac{1}{sin ( x )} 3 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - 2 + \frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )}

Simplify

- 2 + \frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )} : \frac{- 2 cos ( x ) + 3}{cos ( x )}

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

\frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )} = \frac{3}{cos ( x )}

\frac{1}{sin ( x )} 3 \frac{sin ( x )}{cos ( x )}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{1 \cdot sin ( x ) 3}{sin ( x ) cos ( x )}

Cancel the common factor:

sin (x)

= \frac{1 \cdot 3}{cos ( x )}

Multiply:

1 \cdot 3 = 3

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

= - 2 + \frac{3}{cos ( x )}

Convert element to fraction:

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

= - \frac{2 cos ( x )}{cos ( x )} + \frac{3}{cos ( x )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 2 cos ( x ) + 3}{cos ( x )}

= \frac{- 2 cos ( x ) + 3}{cos ( x )}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 - 2 cos (x) = 0

Move

3

to the right side

3 - 2 cos (x) = 0

Subtract

3

from both sides

3 - 2 cos (x) - 3 = 0 - 3

Simplify

- 2 cos (x) = - 3

- 2 cos (x) = - 3

Divide both sides by

- 2

- 2 cos (x) = - 3

Divide both sides by

- 2

\frac{- 2 cos ( x )}{- 2} = \frac{- 3}{- 2}

Simplify

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

General solutions for

cos (x) = \frac{3}{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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

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

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

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cos^2(2x)-1/4 =0

cos^{2} (2 x) - \frac{1}{4} = 0

cosh(x)= 17/15

cosh (x) = \frac{17}{15}

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4 cos (x) + 23 = 0

Frequently Asked Questions (FAQ)

What is the general solution for csc(x)tan(x)=(2sqrt(3))/3 ?
The general solution for csc(x)tan(x)=(2sqrt(3))/3 is x= pi/6+2pin,x=(11pi)/6+2pin