What is the general solution for csc(x)(cos^2(x)+sqrt(2))=0 ?

The general solution for csc(x)(cos^2(x)+sqrt(2))=0 is No Solution for x\in\mathbb{R}

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Popular Trigonometry >

csc(x)(cos^2(x)+sqrt(2))=0

Solution

csc (x) (cos^{2} (x) + 2) = 0

Solution

No Solution for x \in R

Solution steps

csc (x) (cos^{2} (x) + 2) = 0

Solving each part separately

csc (x) = 0 or cos^{2} (x) + 2 = 0

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

cos^{2} (x) + 2 = 0 :

No Solution

cos^{2} (x) + 2 = 0

Solve by substitution

cos^{2} (x) + 2 = 0

Let:

cos (x) = u

u^{2} + 2 = 0

u^{2} + 2 = 0 : u = 42 i, u = - 42 i

u^{2} + 2 = 0

Move

2

to the right side

u^{2} + 2 = 0

Subtract

2

from both sides

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

Simplify

u^{2} = - 2

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

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

u = - 2, u = - - 2

Simplify

- 2 : 42 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= i 2

2 : 42

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 2^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 42

= i 42

Simplify

- - 2 : - 42 i

- - 2

Simplify

- 2 : 42 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= i 2

2 : 42

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{\frac{1}{2}}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot \frac{1}{2}}

\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}

\frac{1}{2} \cdot \frac{1}{2}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 1 = 1

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{1}{4}

= 2^{\frac{1}{4}}

a^{\frac{1}{n}} = n a

= 42

= i 42

= - 42 i

u = 42 i, u = - 42 i

Substitute back

u = cos (x)

cos (x) = 42 i, cos (x) = - 42 i

cos (x) = 42 i, cos (x) = - 42 i

cos (x) = 42 i :

No Solution

cos (x) = 42 i

No Solution

cos (x) = - 42 i :

No Solution

cos (x) = - 42 i

No Solution

Combine all the solutions

No Solution

Combine all the solutions

No Solution for x \in R

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for csc(x)(cos^2(x)+sqrt(2))=0 ?
The general solution for csc(x)(cos^2(x)+sqrt(2))=0 is No Solution for x\in\mathbb{R}