What is the general solution for -4sin(x)=-cos^2(x)+5 ?

The general solution for -4sin(x)=-cos^2(x)+5 is No Solution for x\in\mathbb{R}

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Popular >

-4sin(x)=-cos^2(x)+5

Solution

- 4 sin (x) = - cos^{2} (x) + 5

Solution

No Solution for x \in R

Solution steps

- 4 sin (x) = - cos^{2} (x) + 5

Subtract

- cos^{2} (x) + 5

from both sides

- 4 sin (x) + cos^{2} (x) - 5 = 0

Rewrite using trig identities

- 5 + cos^{2} (x) - 4 sin (x)

Use the Pythagorean identity:

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

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

= - 5 + 1 - sin^{2} (x) - 4 sin (x)

Simplify

= - sin^{2} (x) - 4 sin (x) - 4

- 4 - sin^{2} (x) - 4 sin (x) = 0

Solve by substitution

- 4 - sin^{2} (x) - 4 sin (x) = 0

Let:

sin (x) = u

- 4 - u^{2} - 4 u = 0

- 4 - u^{2} - 4 u = 0 : u = - 2

- 4 - u^{2} - 4 u = 0

Write in the standard form

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

- u^{2} - 4 u - 4 = 0

Solve with the quadratic formula

- u^{2} - 4 u - 4 = 0

Quadratic Equation Formula:

For

a = - 1, b = - 4, c = - 4

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

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

(- 4)^{2} - 4 (- 1) (- 4) = 0

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

(- 4)^{2} = 4^{2}

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

Multiply the numbers:

4 \cdot 1 \cdot 4 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 ( - 1 )}

u = \frac{- ( - 4 )}{2 ( - 1 )}

\frac{- ( - 4 )}{2 ( - 1 )} = - 2

\frac{- ( - 4 )}{2 ( - 1 )}

Remove parentheses:

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

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{4}{- 2}

Apply the fraction rule:

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

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

u = - 2

The solution to the quadratic equation is:

u = - 2

Substitute back

u = sin (x)

sin (x) = - 2

sin (x) = - 2

sin (x) = - 2 :

No Solution

sin (x) = - 2

- 1 \leq sin (x) \leq 1

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 -4sin(x)=-cos^2(x)+5 ?
The general solution for -4sin(x)=-cos^2(x)+5 is No Solution for x\in\mathbb{R}