What is the general solution for 3sin(x)+csc(x)=0 ?

The general solution for 3sin(x)+csc(x)=0 is No Solution for x\in\mathbb{R}

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Popular >

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

Solution

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

Solution

No Solution for x \in R

Solution steps

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

Rewrite using trig identities

csc (x) + 3 sin (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

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

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

Solve by substitution

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

Let:

csc (x) = u

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

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

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

\frac{3}{u} u

Multiply fractions:

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

= \frac{3 u}{u}

Cancel the common factor:

u

= 3

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} + 3 = 0

u^{2} + 3 = 0

u^{2} + 3 = 0

Solve

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

u^{2} + 3 = 0

Move

3

to the right side

u^{2} + 3 = 0

Subtract

3

from both sides

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

Simplify

u^{2} = - 3

u^{2} = - 3

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 3, u = - - 3

Simplify

- 3 : 3 i

- 3

Apply radical rule:

- a = - 1 a

- 3 = - 1 3

= - 1 3

Apply imaginary number rule:

- 1 = i

= 3 i

Simplify

- - 3 : - 3 i

- - 3

Simplify

- 3 : 3 i

- 3

Apply radical rule:

- a = - 1 a

- 3 = - 1 3

= - 1 3

Apply imaginary number rule:

- 1 = i

= 3 i

= - 3 i

u = 3 i, u = - 3 i

u = 3 i, u = - 3 i

Substitute back

u = csc (x)

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

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

csc (x) = 3 i :

No Solution

csc (x) = 3 i

No Solution

csc (x) = - 3 i :

No Solution

csc (x) = - 3 i

No Solution

Combine all the solutions

No Solution for x \in R

Popular Examples

cos(x)= 4/7

cos (x) = \frac{4}{7}

csc^2(x)-2cot(x)=0

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

tan(θ)=-4/3

tan (θ) = - \frac{4}{3}

5csc(θ)-3=2

5 csc (θ) - 3 = 2

3cos(θ-55)=-1.56

3 cos (θ - 5 5^{\circ}) = - 1.56

Frequently Asked Questions (FAQ)

What is the general solution for 3sin(x)+csc(x)=0 ?
The general solution for 3sin(x)+csc(x)=0 is No Solution for x\in\mathbb{R}