What is the general solution for -4csc(x)=-csc^2(x)-4 ?

The general solution for -4csc(x)=-csc^2(x)-4 is x= pi/6+2pin,x=(5pi)/6+2pin

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Popular Trigonometry >

-4csc(x)=-csc^2(x)-4

Solution

- 4 csc (x) = - csc^{2} (x) - 4

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

- 4 csc (x) = - csc^{2} (x) - 4

Solve by substitution

- 4 csc (x) = - csc^{2} (x) - 4

Let:

csc (x) = u

- 4 u = - u^{2} - 4

- 4 u = - u^{2} - 4 : u = 2

- 4 u = - u^{2} - 4

Switch sides

- u^{2} - 4 = - 4 u

Move

4 u

to the left side

- u^{2} - 4 = - 4 u

Add

4 u

to both sides

- u^{2} - 4 + 4 u = - 4 u + 4 u

Simplify

- u^{2} - 4 + 4 u = 0

- u^{2} - 4 + 4 u = 0

Write in the standard form

a x^{2} + b x + c = 0

- u^{2} + 4 u - 4 = 0

Solve with the quadratic formula

- u^{2} + 4 u - 4 = 0

Quadratic Equation Formula:

For

a = - 1, b = 4, c = - 4

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 1 ) ( - 4 )}{2 ( - 1 )}

u_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 ( - 1 ) ( - 4 )}{2 ( - 1 )}

4^{2} - 4 (- 1) (- 4) = 0

4^{2} - 4 (- 1) (- 4)

Apply rule

- (- a) = a

= 4^{2} - 4 \cdot 1 \cdot 4

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 ( - 1 )}

u = \frac{- 4}{2 ( - 1 )}

\frac{- 4}{2 ( - 1 )} = 2

\frac{- 4}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 4}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4}{- 2}

Apply the fraction rule:

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

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

u = 2

The solution to the quadratic equation is:

u = 2

Substitute back

u = csc (x)

csc (x) = 2

csc (x) = 2

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

General solutions for

csc (x) = 2

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}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -4csc(x)=-csc^2(x)-4 ?
The general solution for -4csc(x)=-csc^2(x)-4 is x= pi/6+2pin,x=(5pi)/6+2pin