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

The general solution for (csc(x))/(cot(x))=sqrt(2) is x= pi/4+2pin,x=(7pi)/4+2pin

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(csc(x))/(cot(x))=sqrt(2)

Solution

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

Solution

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Simplify

\frac{csc ( x )}{cot ( x )} - 2 : \frac{csc ( x ) - 2 cot ( x )}{cot ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

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

csc (x) - 2 cot (x) = 0

Express with sin, cos

csc (x) - cot (x) 2

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply

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

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

Multiply fractions:

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

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

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

Apply rule

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

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

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

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

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

1 - cos (x) 2 = 0

Move

1

to the right side

1 - cos (x) 2 = 0

Subtract

1

from both sides

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

Simplify

- cos (x) 2 = - 1

- cos (x) 2 = - 1

Divide both sides by

- 2

- cos (x) 2 = - 1

Divide both sides by

- 2

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= cos (x)

Simplify

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

Apply the fraction rule:

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

= \frac{1}{2}

Rationalize

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiply by the conjugate

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Apply radical rule:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

General solutions for

cos (x) = \frac{2}{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}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Graph

Popular Examples

sin(4x)+6cos(2x)=0

sin (4 x) + 6 cos (2 x) = 0

sec(θ)+tan(θ)=0

sec (θ) + tan (θ) = 0

sin(θ)=(3.5)/5

sin (θ) = \frac{3.5}{5}

sec(a)= 41/12 ,2pi<a<(5pi)/2 ,sin(a/2)

sec (a) = \frac{41}{12}, 2 π < a < \frac{5 π}{2}, sin (\frac{a}{2})

sin(2x)=0.3

sin (2 x) = 0.3

Frequently Asked Questions (FAQ)

What is the general solution for (csc(x))/(cot(x))=sqrt(2) ?
The general solution for (csc(x))/(cot(x))=sqrt(2) is x= pi/4+2pin,x=(7pi)/4+2pin