What is the general solution for 2tan(x/2)-csc(x)=0 ?

The general solution for 2tan(x/2)-csc(x)=0 is x= pi/3+(4pin)/3 ,x=pi+(4pin)/3

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Popular Trigonometry >

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

Solution

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

Solution

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

Degrees

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n

Solution steps

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

2 \cdot \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )} : \frac{2 sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

2 \cdot \frac{sin ( \frac{x}{2} )}{cos ( \frac{x}{2} )}

Multiply fractions:

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

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

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

Least Common Multiplier of

sin (x), cos (\frac{x}{2}) : sin (x) cos (\frac{x}{2})

sin (x), cos (\frac{x}{2})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

cos (\frac{x}{2})

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

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

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

For

\frac{1}{sin ( x )} :

multiply the denominator and numerator by

cos (\frac{x}{2})

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

For

\frac{sin ( \frac{x}{2} ) \cdot 2}{cos ( \frac{x}{2} )} :

multiply the denominator and numerator by

sin (x)

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Use the Product to Sum identity:

sin (s) sin (t) = \frac{1}{2} (cos (s - t) - cos (s + t))

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Join

\frac{x}{2} - x : - \frac{x}{2}

\frac{x}{2} - x

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{x}{2} - \frac{x \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{x - x \cdot 2}{2}

Add similar elements:

x - 2 x = - x

= \frac{- x}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{x}{2}

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

Join

\frac{x}{2} + x : \frac{3 x}{2}

\frac{x}{2} + x

Convert element to fraction:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{x + x \cdot 2}{2}

Add similar elements:

x + 2 x = 3 x

= \frac{3 x}{2}

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

Refine

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

Use the negative angle identity:

cos (- x) = cos (x)

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

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

Add similar elements:

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

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

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

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

General solutions for

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

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

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

Solve

\frac{3 x}{2} = \frac{π}{2} + 2 πn : x = \frac{π}{3} + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

Divide both sides by

3

3 x = π + 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{π}{3} + \frac{4 πn}{3}

Simplify

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

Solve

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn : x = π + \frac{4 πn}{3}

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

Divide both sides by

3

3 x = 3 π + 4 πn

Divide both sides by

3

\frac{3 x}{3} = \frac{3 π}{3} + \frac{4 πn}{3}

Simplify

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

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Frequently Asked Questions (FAQ)

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