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

The general solution for sin(x)cot(x)=-(sqrt(2))/2 is x=(3pi)/4+2pin,x=(5pi)/4+2pin

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Popular Trigonometry >

sin(x)cot(x)=-(sqrt(2))/2

Solution

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

Solution

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

- \frac{2}{2}

from both sides

sin (x) cot (x) + \frac{1}{2} = 0

Simplify

sin (x) cot (x) + \frac{1}{2} : \frac{2 sin ( x ) cot ( x ) + 1}{2}

sin (x) cot (x) + \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

\frac{2 sin ( x ) cot ( x ) + 1}{2} = 0

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

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

Rewrite using trig identities

1 + cot (x) sin (x) 2

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= cos (x) 2

= 1 + 2 cos (x)

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}

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{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)cot(x)=-(sqrt(2))/2 ?
The general solution for sin(x)cot(x)=-(sqrt(2))/2 is x=(3pi)/4+2pin,x=(5pi)/4+2pin