What is the general solution for csc(x)+sin(x)=-3 ?

The general solution for csc(x)+sin(x)=-3 is x=-0.39192…+2pin,x=pi+0.39192…+2pin

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Popular >

csc(x)+sin(x)=-3

Solution

csc (x) + sin (x) = - 3

Solution

x = - 0.39192 \dots + 2 πn, x = π + 0.39192 \dots + 2 πn

Degrees

x = - 22.45551 \dots^{\circ} + 36 0^{\circ} n, x = 202.45551 \dots^{\circ} + 36 0^{\circ} n

Solution steps

csc (x) + sin (x) = - 3

Subtract

- 3

from both sides

csc (x) + sin (x) + 3 = 0

Rewrite using trig identities

3 + csc (x) + sin (x)

Use the basic trigonometric identity:

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

= 3 + csc (x) + \frac{1}{csc ( x )}

3 + csc (x) + \frac{1}{csc ( x )} = 0

Solve by substitution

3 + csc (x) + \frac{1}{csc ( x )} = 0

Let:

csc (x) = u

3 + u + \frac{1}{u} = 0

3 + u + \frac{1}{u} = 0 : u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

3 + u + \frac{1}{u} = 0

Multiply both sides by

u

3 + u + \frac{1}{u} = 0

Multiply both sides by

u

3 u + uu + \frac{1}{u} u = 0 \cdot u

Simplify

3 u + uu + \frac{1}{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{1}{u} u : 1

\frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot u}{u}

Cancel the common factor:

u

= 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

3 u + u^{2} + 1 = 0

3 u + u^{2} + 1 = 0

3 u + u^{2} + 1 = 0

Solve

3 u + u^{2} + 1 = 0 : u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

3 u + u^{2} + 1 = 0

Write in the standard form

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

u^{2} + 3 u + 1 = 0

Solve with the quadratic formula

u^{2} + 3 u + 1 = 0

Quadratic Equation Formula:

For

a = 1, b = 3, c = 1

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

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

3^{2} - 4 \cdot 1 \cdot 1 = 5

3^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

Subtract the numbers:

9 - 4 = 5

= 5

u_{1, 2} = \frac{- 3 \pm 5}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 3 + 5}{2 \cdot 1}, u_{2} = \frac{- 3 - 5}{2 \cdot 1}

u = \frac{- 3 + 5}{2 \cdot 1} : \frac{- 3 + 5}{2}

\frac{- 3 + 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 + 5}{2}

u = \frac{- 3 - 5}{2 \cdot 1} : \frac{- 3 - 5}{2}

\frac{- 3 - 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 3 - 5}{2}

The solutions to the quadratic equation are:

u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

3 + u + \frac{1}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{- 3 + 5}{2}, u = \frac{- 3 - 5}{2}

Substitute back

u = csc (x)

csc (x) = \frac{- 3 + 5}{2}, csc (x) = \frac{- 3 - 5}{2}

csc (x) = \frac{- 3 + 5}{2}, csc (x) = \frac{- 3 - 5}{2}

csc (x) = \frac{- 3 + 5}{2} :

No Solution

csc (x) = \frac{- 3 + 5}{2}

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

No Solution

csc (x) = \frac{- 3 - 5}{2} : x = arccsc (\frac{- 3 - 5}{2}) + 2 πn, x = π + arccsc (- \frac{- 3 - 5}{2}) + 2 πn

csc (x) = \frac{- 3 - 5}{2}

Apply trig inverse properties

csc (x) = \frac{- 3 - 5}{2}

General solutions for

csc (x) = \frac{- 3 - 5}{2}

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

x = arccsc (\frac{- 3 - 5}{2}) + 2 πn, x = π + arccsc (- \frac{- 3 - 5}{2}) + 2 πn

x = arccsc (\frac{- 3 - 5}{2}) + 2 πn, x = π + arccsc (- \frac{- 3 - 5}{2}) + 2 πn

Combine all the solutions

x = arccsc (\frac{- 3 - 5}{2}) + 2 πn, x = π + arccsc (- \frac{- 3 - 5}{2}) + 2 πn

Show solutions in decimal form

x = - 0.39192 \dots + 2 πn, x = π + 0.39192 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)+sin(x)=-3 ?
The general solution for csc(x)+sin(x)=-3 is x=-0.39192…+2pin,x=pi+0.39192…+2pin