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

The general solution for 2sin(x)-csc(x)=2 is x=-0.37473…+2pin,x=pi+0.37473…+2pin

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2sin(x)-csc(x)=2

Solution

2 sin (x) - csc (x) = 2

Solution

x = - 0.37473 \dots + 2 πn, x = π + 0.37473 \dots + 2 πn

Degrees

x = - 21.47070 \dots^{\circ} + 36 0^{\circ} n, x = 201.47070 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 sin (x) - csc (x) = 2

Subtract

2

from both sides

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

= - 2 - 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 )}

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

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

Solve by substitution

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

Let:

csc (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

- 2 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

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

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

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

Solve

- 2 u - u^{2} + 2 = 0 : u = - 1 - 3, u = 3 - 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 2)^{2} - 4 (- 1) \cdot 2 = 23

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

Add the numbers:

4 + 8 = 12

= 12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

= 3 2^{2}

Apply radical rule:

2^{2} = 2

= 23

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

Separate the solutions

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

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

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

Remove parentheses:

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

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

Cancel

\frac{2 + 2 3}{2} : 1 + 3

\frac{2 + 2 3}{2}

Factor

2 + 23 : 2 (1 + 3)

2 + 23

Rewrite as

= 2 \cdot 1 + 23

Factor out common term

2

= 2 (1 + 3)

= \frac{2 ( 1 + 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 3

= - (1 + 3)

Distribute parentheses

= - (1) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 3

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

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

Remove parentheses:

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

2 - 23 = - (23 - 2)

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

Factor

23 - 2 : 2 (3 - 1)

23 - 2

Rewrite as

= 23 - 2 \cdot 1

Factor out common term

2

= 2 (3 - 1)

= \frac{2 ( 3 - 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 - 1

The solutions to the quadratic equation are:

u = - 1 - 3, u = 3 - 1

u = - 1 - 3, u = 3 - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

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

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1 - 3, u = 3 - 1

Substitute back

u = csc (x)

csc (x) = - 1 - 3, csc (x) = 3 - 1

csc (x) = - 1 - 3, csc (x) = 3 - 1

csc (x) = - 1 - 3 : x = arccsc (- 1 - 3) + 2 πn, x = π + arccsc (1 + 3) + 2 πn

csc (x) = - 1 - 3

Apply trig inverse properties

csc (x) = - 1 - 3

General solutions for

csc (x) = - 1 - 3

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

x = arccsc (- 1 - 3) + 2 πn, x = π + arccsc (1 + 3) + 2 πn

x = arccsc (- 1 - 3) + 2 πn, x = π + arccsc (1 + 3) + 2 πn

csc (x) = 3 - 1 :

No Solution

csc (x) = 3 - 1

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

No Solution

Combine all the solutions

x = arccsc (- 1 - 3) + 2 πn, x = π + arccsc (1 + 3) + 2 πn

Show solutions in decimal form

x = - 0.37473 \dots + 2 πn, x = π + 0.37473 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(x)-csc(x)=2 ?
The general solution for 2sin(x)-csc(x)=2 is x=-0.37473…+2pin,x=pi+0.37473…+2pin