What is the general solution for sin^2(x)+7sin^2(x)+5cos^2(x)+3=0 ?

The general solution for sin^2(x)+7sin^2(x)+5cos^2(x)+3=0 is No Solution for x\in\mathbb{R}

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sin^2(x)+7sin^2(x)+5cos^2(x)+3=0

Solution

sin^{2} (x) + 7 sin^{2} (x) + 5 cos^{2} (x) + 3 = 0

Solution

No Solution for x \in R

Solution steps

sin^{2} (x) + 7 sin^{2} (x) + 5 cos^{2} (x) + 3 = 0

Rewrite using trig identities

3 + sin^{2} (x) + 5 cos^{2} (x) + 7 sin^{2} (x)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= 3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x)

Simplify

3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x) : 3 sin^{2} (x) + 8

3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) + 7 sin^{2} (x)

Expand

5 (1 - sin^{2} (x)) : 5 - 5 sin^{2} (x)

5 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

= 5 \cdot 1 - 5 sin^{2} (x)

Multiply the numbers:

5 \cdot 1 = 5

= 5 - 5 sin^{2} (x)

= 3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x)

Simplify

3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x) : 3 sin^{2} (x) + 8

3 + sin^{2} (x) + 5 - 5 sin^{2} (x) + 7 sin^{2} (x)

Group like terms

= sin^{2} (x) - 5 sin^{2} (x) + 7 sin^{2} (x) + 3 + 5

Add similar elements:

sin^{2} (x) - 5 sin^{2} (x) + 7 sin^{2} (x) = 3 sin^{2} (x)

= 3 sin^{2} (x) + 3 + 5

Add the numbers:

3 + 5 = 8

= 3 sin^{2} (x) + 8

= 3 sin^{2} (x) + 8

= 3 sin^{2} (x) + 8

8 + 3 sin^{2} (x) = 0

Solve by substitution

8 + 3 sin^{2} (x) = 0

Let:

sin (x) = u

8 + 3 u^{2} = 0

8 + 3 u^{2} = 0 : u = i \frac{2 6}{3}, u = - i \frac{2 6}{3}

8 + 3 u^{2} = 0

Move

8

to the right side

8 + 3 u^{2} = 0

Subtract

8

from both sides

8 + 3 u^{2} - 8 = 0 - 8

Simplify

3 u^{2} = - 8

3 u^{2} = - 8

Divide both sides by

3

3 u^{2} = - 8

Divide both sides by

3

\frac{3 u ^{2}}{3} = \frac{- 8}{3}

Simplify

u^{2} = - \frac{8}{3}

u^{2} = - \frac{8}{3}

For

x^{2} = f (a)

the solutions are

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

u = - \frac{8}{3}, u = - - \frac{8}{3}

Simplify

- \frac{8}{3} : i \frac{2 6}{3}

- \frac{8}{3}

Apply radical rule:

- a = - 1 a

- \frac{8}{3} = - 1 \frac{8}{3}

= - 1 \frac{8}{3}

Apply imaginary number rule:

- 1 = i

= i \frac{8}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{8}{3} = \frac{8}{3}

= i \frac{8}{3}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{2 2}{3}

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

Multiply by the conjugate

\frac{3}{3}

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

223 = 26

223

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 26

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= i \frac{2 6}{3}

Rewrite

i \frac{2 6}{3}

in standard complex form:

\frac{2 6}{3} i

i \frac{2 6}{3}

\frac{2 6}{3} = \frac{2 2}{3}

\frac{2 6}{3}

Factor

6 : 23

Factor

6 = 2 \cdot 3

= 2 \cdot 3

Apply radical rule:

= 23

= \frac{2 2 3}{3}

Cancel

\frac{2 2 3}{3} : \frac{2 2}{3}

\frac{2 2 3}{3}

Apply radical rule:

3 = 3^{\frac{1}{2}}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

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

= \frac{2 2}{3 ^{\frac{1}{2}}}

Apply radical rule:

3^{\frac{1}{2}} = 3

= \frac{2 2}{3}

= \frac{2 2}{3}

= i \frac{2 2}{3}

Multiply fractions:

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

= \frac{2 2 i}{3}

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

Multiply by the conjugate

\frac{3}{3}

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

223 = 26

223

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 26

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= \frac{2 6}{3} i

= \frac{2 6}{3} i

Simplify

- - \frac{8}{3} : - i \frac{2 6}{3}

- - \frac{8}{3}

Simplify

- \frac{8}{3} : i \frac{2 2}{3}

- \frac{8}{3}

Apply radical rule:

- a = - 1 a

- \frac{8}{3} = - 1 \frac{8}{3}

= - 1 \frac{8}{3}

Apply imaginary number rule:

- 1 = i

= i \frac{8}{3}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{8}{3} = \frac{8}{3}

= i \frac{8}{3}

8 = 22

8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= i \frac{2 2}{3}

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

\frac{2 2}{3} = \frac{2 6}{3}

\frac{2 2}{3}

Multiply by the conjugate

\frac{3}{3}

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

223 = 26

223

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 26

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{2 6}{3}

= - \frac{2 6}{3} i

u = i \frac{2 6}{3}, u = - i \frac{2 6}{3}

Substitute back

u = sin (x)

sin (x) = i \frac{2 6}{3}, sin (x) = - i \frac{2 6}{3}

sin (x) = i \frac{2 6}{3}, sin (x) = - i \frac{2 6}{3}

sin (x) = i \frac{2 6}{3} :

No Solution

sin (x) = i \frac{2 6}{3}

No Solution

sin (x) = - i \frac{2 6}{3} :

No Solution

sin (x) = - i \frac{2 6}{3}

No Solution

Combine all the solutions

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)+7sin^2(x)+5cos^2(x)+3=0 ?
The general solution for sin^2(x)+7sin^2(x)+5cos^2(x)+3=0 is No Solution for x\in\mathbb{R}