What is the general solution for 5+cot^2(x)=4csc(x) ?

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

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5+cot^2(x)=4csc(x)

Solution

5 + cot^{2} (x) = 4 csc (x)

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

5 + cot^{2} (x) = 4 csc (x)

Subtract

4 csc (x)

from both sides

5 + cot^{2} (x) - 4 csc (x) = 0

Rewrite using trig identities

5 + cot^{2} (x) - 4 csc (x)

Use the Pythagorean identity:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= 5 + csc^{2} (x) - 1 - 4 csc (x)

Simplify

5 + csc^{2} (x) - 1 - 4 csc (x) : csc^{2} (x) - 4 csc (x) + 4

5 + csc^{2} (x) - 1 - 4 csc (x)

Group like terms

= csc^{2} (x) - 4 csc (x) + 5 - 1

Add/Subtract the numbers:

5 - 1 = 4

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

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

4 + csc^{2} (x) - 4 csc (x) = 0

Solve by substitution

4 + csc^{2} (x) - 4 csc (x) = 0

Let:

csc (x) = u

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

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

4 + u^{2} - 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 \cdot 1 \cdot 4}{2 \cdot 1}

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

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

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

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

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

= 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 \cdot 1}

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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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