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

The general solution for -sec(x/2)=2csc(x/2) is x=-2*1.10714…+2pin

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Popular >

-sec(x/2)=2csc(x/2)

Solution

- sec (\frac{x}{2}) = 2 csc (\frac{x}{2})

Solution

x = - 2 \cdot 1.10714 \dots + 2 πn

Degrees

x = - 126.86989 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- sec (\frac{x}{2}) = 2 csc (\frac{x}{2})

Subtract

2 csc (\frac{x}{2})

from both sides

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

Express with sin, cos

- sec (\frac{x}{2}) - 2 csc (\frac{x}{2})

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

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

Least Common Multiplier of

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

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

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

cos (\frac{x}{2})

sin (\frac{x}{2})

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

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

For

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

multiply the denominator and numerator by

sin (\frac{x}{2})

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

For

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

multiply the denominator and numerator by

cos (\frac{x}{2})

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

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

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

Rewrite using trig identities

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

Divide both sides by

cos (\frac{x}{2}), cos (\frac{x}{2}) \neq = 0

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

Simplify

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

Use the basic trigonometric identity:

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

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

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

Move

2

to the right side

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

Add

2

to both sides

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

Simplify

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

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

Divide both sides by

- 1

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

Divide both sides by

- 1

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

Simplify

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

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = - a \Rightarrow x = arctan (- a) + πn

\frac{x}{2} = arctan (- 2) + πn

\frac{x}{2} = arctan (- 2) + πn

Solve

\frac{x}{2} = arctan (- 2) + πn : x = - 2 arctan (2) + 2 πn

\frac{x}{2} = arctan (- 2) + πn

Simplify

arctan (- 2) + πn : - arctan (2) + πn

arctan (- 2) + πn

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

= - arctan (2) + πn

\frac{x}{2} = - arctan (2) + πn

Multiply both sides by

2

\frac{x}{2} = - arctan (2) + πn

Multiply both sides by

2

\frac{2 x}{2} = - 2 arctan (2) + 2 πn

Simplify

x = - 2 arctan (2) + 2 πn

x = - 2 arctan (2) + 2 πn

x = - 2 arctan (2) + 2 πn

Show solutions in decimal form

x = - 2 \cdot 1.10714 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for -sec(x/2)=2csc(x/2) ?
The general solution for -sec(x/2)=2csc(x/2) is x=-2*1.10714…+2pin