What is the general solution for 5-3csc^2(x)=1 ?

The general solution for 5-3csc^2(x)=1 is x= pi/3+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=(5pi)/3+2pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

5-3csc^2(x)=1

Solution

5 - 3 csc^{2} (x) = 1

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 - 3 csc^{2} (x) = 1

Solve by substitution

5 - 3 csc^{2} (x) = 1

Let:

csc (x) = u

5 - 3 u^{2} = 1

5 - 3 u^{2} = 1 : u = \frac{2 3}{3}, u = - \frac{2 3}{3}

5 - 3 u^{2} = 1

Move

5

to the right side

5 - 3 u^{2} = 1

Subtract

5

from both sides

5 - 3 u^{2} - 5 = 1 - 5

Simplify

- 3 u^{2} = - 4

- 3 u^{2} = - 4

Divide both sides by

- 3

- 3 u^{2} = - 4

Divide both sides by

- 3

\frac{- 3 u ^{2}}{- 3} = \frac{- 4}{- 3}

Simplify

u^{2} = \frac{4}{3}

u^{2} = \frac{4}{3}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{4}{3}, u = - \frac{4}{3}

\frac{4}{3} = \frac{2 3}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

Rationalize

\frac{2}{3} : \frac{2 3}{3}

\frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 3}{3}

= \frac{2 3}{3}

- \frac{4}{3} = - \frac{2 3}{3}

- \frac{4}{3}

Simplify

\frac{4}{3} : \frac{2}{3}

\frac{4}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{4}{3}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{2}{3}

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

= - \frac{2 3}{3 3}

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

u = \frac{2 3}{3}, u = - \frac{2 3}{3}

Substitute back

u = csc (x)

csc (x) = \frac{2 3}{3}, csc (x) = - \frac{2 3}{3}

csc (x) = \frac{2 3}{3}, csc (x) = - \frac{2 3}{3}

csc (x) = \frac{2 3}{3} : x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

csc (x) = \frac{2 3}{3}

General solutions for

csc (x) = \frac{2 3}{3}

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

csc (x) = - \frac{2 3}{3} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

csc (x) = - \frac{2 3}{3}

General solutions for

csc (x) = - \frac{2 3}{3}

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

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

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

Combine all the solutions

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

Popular Examples

-1-3cos^2(x)+2cos(x)sqrt(3)=0

- 1 - 3 cos^{2} (x) + 2 cos (x) 3 = 0

tan(3x)+cos(6x)=1

tan (3 x) + cos (6 x) = 1

sec(θ)+tan(θ)=2

sec (θ) + tan (θ) = 2

sin^2(θ)=2cos(θ)+2,0<= θ<= 2pi

sin^{2} (θ) = 2 cos (θ) + 2, 0 \leq θ \leq 2 π

sin(x)=-0.3

sin (x) = - 0.3

Frequently Asked Questions (FAQ)

What is the general solution for 5-3csc^2(x)=1 ?
The general solution for 5-3csc^2(x)=1 is x= pi/3+2pin,x=(2pi)/3+2pin,x=(4pi)/3+2pin,x=(5pi)/3+2pin