What is the general solution for 3cot^2(x)=7csc(x)-7 ?

The general solution for 3cot^2(x)=7csc(x)-7 is x=0.84806…+2pin,x=pi-0.84806…+2pin,x= pi/2+2pin

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3cot^2(x)=7csc(x)-7

Solution

3 cot^{2} (x) = 7 csc (x) - 7

Solution

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

3 cot^{2} (x) = 7 csc (x) - 7

Subtract

7 csc (x) - 7

from both sides

3 cot^{2} (x) - 7 csc (x) + 7 = 0

Rewrite using trig identities

7 + 3 cot^{2} (x) - 7 csc (x)

Use the Pythagorean identity:

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

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

= 7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

Simplify

7 + 3 (csc^{2} (x) - 1) - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 (csc^{2} (x) - 1) - 7 csc (x)

Expand

3 (csc^{2} (x) - 1) : 3 csc^{2} (x) - 3

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

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = csc^{2} (x), c = 1

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

Multiply the numbers:

3 \cdot 1 = 3

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

= 7 + 3 csc^{2} (x) - 3 - 7 csc (x)

Simplify

7 + 3 csc^{2} (x) - 3 - 7 csc (x) : 3 csc^{2} (x) - 7 csc (x) + 4

7 + 3 csc^{2} (x) - 3 - 7 csc (x)

Group like terms

= 3 csc^{2} (x) - 7 csc (x) + 7 - 3

Add/Subtract the numbers:

7 - 3 = 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

= 3 csc^{2} (x) - 7 csc (x) + 4

4 + 3 csc^{2} (x) - 7 csc (x) = 0

Solve by substitution

4 + 3 csc^{2} (x) - 7 csc (x) = 0

Let:

csc (x) = u

4 + 3 u^{2} - 7 u = 0

4 + 3 u^{2} - 7 u = 0 : u = \frac{4}{3}, u = 1

4 + 3 u^{2} - 7 u = 0

Write in the standard form

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

3 u^{2} - 7 u + 4 = 0

Solve with the quadratic formula

3 u^{2} - 7 u + 4 = 0

Quadratic Equation Formula:

For

a = 3, b = - 7, c = 4

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 3 \cdot 4}{2 \cdot 3}

(- 7)^{2} - 4 \cdot 3 \cdot 4 = 1

(- 7)^{2} - 4 \cdot 3 \cdot 4

Apply exponent rule:

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

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 7^{2} - 48

7^{2} = 49

= 49 - 48

Subtract the numbers:

49 - 48 = 1

= 1

Apply rule

1 = 1

= 1

u_{1, 2} = \frac{- ( - 7 ) \pm 1}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 1}{2 \cdot 3}, u_{2} = \frac{- ( - 7 ) - 1}{2 \cdot 3}

u = \frac{- ( - 7 ) + 1}{2 \cdot 3} : \frac{4}{3}

\frac{- ( - 7 ) + 1}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 + 1}{2 \cdot 3}

Add the numbers:

7 + 1 = 8

= \frac{8}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{8}{6}

Cancel the common factor:

2

= \frac{4}{3}

u = \frac{- ( - 7 ) - 1}{2 \cdot 3} : 1

\frac{- ( - 7 ) - 1}{2 \cdot 3}

Apply rule

- (- a) = a

= \frac{7 - 1}{2 \cdot 3}

Subtract the numbers:

7 - 1 = 6

= \frac{6}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = \frac{4}{3}, u = 1

Substitute back

u = csc (x)

csc (x) = \frac{4}{3}, csc (x) = 1

csc (x) = \frac{4}{3}, csc (x) = 1

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

csc (x) = \frac{4}{3}

Apply trig inverse properties

csc (x) = \frac{4}{3}

General solutions for

csc (x) = \frac{4}{3}

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π - arccsc (a) + 2 πn

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

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

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = 1

General solutions for

csc (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn, x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3cot^2(x)=7csc(x)-7 ?
The general solution for 3cot^2(x)=7csc(x)-7 is x=0.84806…+2pin,x=pi-0.84806…+2pin,x= pi/2+2pin