What is the general solution for 1/(sin(x))=sqrt(5) ?

The general solution for 1/(sin(x))=sqrt(5) is x=0.46364…+2pin,x=pi-0.46364…+2pin

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Popular >

1/(sin(x))=sqrt(5)

Solution

\frac{1}{sin ( x )} = 5

Solution

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

Degrees

x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{1}{sin ( x )} = 5

Multiply both sides by

sin (x)

\frac{1}{sin ( x )} = 5

Multiply both sides by

sin (x)

\frac{1}{sin ( x )} sin (x) = 5 sin (x)

Simplify

1 = 5 sin (x)

1 = 5 sin (x)

Switch sides

5 sin (x) = 1

Divide both sides by

5

5 sin (x) = 1

Divide both sides by

5

\frac{5 sin ( x )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( x )}{5} = \frac{1}{5}

Simplify

\frac{5 sin ( x )}{5} : sin (x)

\frac{5 sin ( x )}{5}

Cancel the common factor:

5

= sin (x)

Simplify

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiply by the conjugate

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Apply radical rule:

a a = a

55 = 5

= 5

= \frac{5}{5}

sin (x) = \frac{5}{5}

sin (x) = \frac{5}{5}

sin (x) = \frac{5}{5}

Verify Solutions

Find undefined (singularity) points:

sin (x) = 0

Take the denominator(s) of

\frac{1}{sin ( x )}

and compare to zero

sin (x) = 0

The following points are undefined

sin (x) = 0

Combine undefined points with solutions:

sin (x) = \frac{5}{5}

Apply trig inverse properties

sin (x) = \frac{5}{5}

General solutions for

sin (x) = \frac{5}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

x = arcsin (\frac{5}{5}) + 2 πn, x = π - arcsin (\frac{5}{5}) + 2 πn

Show solutions in decimal form

x = 0.46364 \dots + 2 πn, x = π - 0.46364 \dots + 2 πn

Popular Examples

tan(2x)+tan(4x)=0

tan (2 x) + tan (4 x) = 0

2sin(θ)-sqrt(3)=0,0<= θ<2pi

2 sin (θ) - 3 = 0, 0 \leq θ < 2 π

arctan(θ)=-1

arctan (θ) = - 1

tan^2(θ)-tan(θ)-2=0

tan^{2} (θ) - tan (θ) - 2 = 0

sin(x)=cos^2(x)-1

sin (x) = cos^{2} (x) - 1

Frequently Asked Questions (FAQ)

What is the general solution for 1/(sin(x))=sqrt(5) ?
The general solution for 1/(sin(x))=sqrt(5) is x=0.46364…+2pin,x=pi-0.46364…+2pin