What is the general solution for 2sin(x)=csc(x)-6,0<= x<2pi ?

The general solution for 2sin(x)=csc(x)-6,0<= x<2pi is x=0.15898…,x=pi-0.15898…

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Popular Trigonometry >

2sin(x)=csc(x)-6,0<= x<2pi

Solution

2 sin (x) = csc (x) - 6, 0 \leq x < 2 π

Solution

x = 0.15898 \dots, x = π - 0.15898 \dots

Degrees

x = 9.10895 \dots^{\circ}, x = 170.89104 \dots^{\circ}

Solution steps

2 sin (x) = csc (x) - 6, 0 \leq x < 2 π

Subtract

csc (x) - 6

from both sides

2 sin (x) - csc (x) + 6 = 0

Rewrite using trig identities

6 - csc (x) + 2 sin (x)

Use the basic trigonometric identity:

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

= 6 - csc (x) + 2 \cdot \frac{1}{csc ( x )}

2 \cdot \frac{1}{csc ( x )} = \frac{2}{csc ( x )}

2 \cdot \frac{1}{csc ( x )}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{csc ( x )}

Multiply the numbers:

1 \cdot 2 = 2

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

= 6 - csc (x) + \frac{2}{csc ( x )}

6 - csc (x) + \frac{2}{csc ( x )} = 0

Solve by substitution

6 - csc (x) + \frac{2}{csc ( x )} = 0

Let:

csc (x) = u

6 - u + \frac{2}{u} = 0

6 - u + \frac{2}{u} = 0 : u = 3 - 11, u = 3 + 11

6 - u + \frac{2}{u} = 0

Multiply both sides by

u

6 - u + \frac{2}{u} = 0

Multiply both sides by

u

6 u - uu + \frac{2}{u} u = 0 \cdot u

Simplify

6 u - uu + \frac{2}{u} u = 0 \cdot u

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

\frac{2}{u} u : 2

\frac{2}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{2 u}{u}

Cancel the common factor:

u

= 2

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

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

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

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

Solve

6 u - u^{2} + 2 = 0 : u = 3 - 11, u = 3 + 11

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

Write in the standard form

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

- u^{2} + 6 u + 2 = 0

Solve with the quadratic formula

- u^{2} + 6 u + 2 = 0

Quadratic Equation Formula:

For

a = - 1, b = 6, c = 2

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 ( - 1 ) \cdot 2}{2 ( - 1 )}

6^{2} - 4 (- 1) \cdot 2 = 211

6^{2} - 4 (- 1) \cdot 2

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 1 \cdot 2

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 6^{2} + 8

6^{2} = 36

= 36 + 8

Add the numbers:

36 + 8 = 44

= 44

Prime factorization of

44 : 2^{2} \cdot 11

44

44

divides by

244 = 22 \cdot 2

= 2 \cdot 22

22

divides by

222 = 11 \cdot 2

= 2 \cdot 2 \cdot 11

2, 11

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 11

= 2^{2} \cdot 11

= 2^{2} \cdot 11

Apply radical rule:

= 11 2^{2}

Apply radical rule:

2^{2} = 2

= 211

u_{1, 2} = \frac{- 6 \pm 2 11}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- 6 + 2 11}{2 ( - 1 )}, u_{2} = \frac{- 6 - 2 11}{2 ( - 1 )}

u = \frac{- 6 + 2 11}{2 ( - 1 )} : 3 - 11

\frac{- 6 + 2 11}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 + 2 11}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6 + 2 11}{- 2}

Apply the fraction rule:

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

= - \frac{- 6 + 2 11}{2}

Cancel

\frac{- 6 + 2 11}{2} : 11 - 3

\frac{- 6 + 2 11}{2}

Factor

- 6 + 211 : 2 (- 3 + 11)

- 6 + 211

Rewrite as

= - 2 \cdot 3 + 211

Factor out common term

2

= 2 (- 3 + 11)

= \frac{2 ( - 3 + 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 3 + 11

= - (11 - 3)

Distribute parentheses

= - (- 3) - (11)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= 3 - 11

u = \frac{- 6 - 2 11}{2 ( - 1 )} : 3 + 11

\frac{- 6 - 2 11}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

= \frac{- 6 - 2 11}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6 - 2 11}{- 2}

Apply the fraction rule:

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

- 6 - 211 = - (6 + 211)

= \frac{6 + 2 11}{2}

Factor

6 + 211 : 2 (3 + 11)

6 + 211

Rewrite as

= 2 \cdot 3 + 211

Factor out common term

2

= 2 (3 + 11)

= \frac{2 ( 3 + 11 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 + 11

The solutions to the quadratic equation are:

u = 3 - 11, u = 3 + 11

u = 3 - 11, u = 3 + 11

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

6 - u + \frac{2}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 3 - 11, u = 3 + 11

Substitute back

u = csc (x)

csc (x) = 3 - 11, csc (x) = 3 + 11

csc (x) = 3 - 11, csc (x) = 3 + 11

csc (x) = 3 - 11, 0 \leq x < 2 π :

No Solution

csc (x) = 3 - 11, 0 \leq x < 2 π

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

csc (x) = 3 + 11, 0 \leq x < 2 π : x = arccsc (3 + 11), x = π - arccsc (3 + 11)

csc (x) = 3 + 11, 0 \leq x < 2 π

Apply trig inverse properties

csc (x) = 3 + 11

General solutions for

csc (x) = 3 + 11

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

x = arccsc (3 + 11) + 2 πn, x = π - arccsc (3 + 11) + 2 πn

x = arccsc (3 + 11) + 2 πn, x = π - arccsc (3 + 11) + 2 πn

Solutions for the range

0 \leq x < 2 π

x = arccsc (3 + 11), x = π - arccsc (3 + 11)

Combine all the solutions

x = arccsc (3 + 11), x = π - arccsc (3 + 11)

Show solutions in decimal form

x = 0.15898 \dots, x = π - 0.15898 \dots

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)=csc(x)-6,0<= x<2pi ?
The general solution for 2sin(x)=csc(x)-6,0<= x<2pi is x=0.15898…,x=pi-0.15898…