What is the general solution for sin^2(x)csc(x)=-1/2 ?

The general solution for sin^2(x)csc(x)=-1/2 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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sin^2(x)csc(x)=-1/2

Solution

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

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

- \frac{1}{2}

from both sides

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

Simplify

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

sin^{2} (x) csc (x) + \frac{1}{2}

Convert element to fraction:

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

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

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

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

2 sin^{2} (x) csc (x) + 1 = 0

Rewrite using trig identities

1 + 2 csc (x) sin^{2} (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

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

Cancel the common factor:

sin (x)

= 2 sin (x)

= 1 + 2 sin (x)

1 + 2 sin (x) = 0

Move

1

to the right side

1 + 2 sin (x) = 0

Subtract

1

from both sides

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

Simplify

2 sin (x) = - 1

2 sin (x) = - 1

Divide both sides by

2

2 sin (x) = - 1

Divide both sides by

2

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

Simplify

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

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

General solutions for

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

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

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Popular Examples

sec^2(2x)-1=0

sec^{2} (2 x) - 1 = 0

1=2cos(x)

1 = 2 cos (x)

solvefor x,y=arcsin(x/(11))solve for

x, y = arcsin (\frac{x}{11})

(sin(48))/(13)=(sin(b))/(16)

\frac{sin ( 4 8 ^{\circ} )}{13} = \frac{sin ( b )}{16}

9cos(2θ)=0

9 cos (2 θ) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)csc(x)=-1/2 ?
The general solution for sin^2(x)csc(x)=-1/2 is x=(7pi)/6+2pin,x=(11pi)/6+2pin