What is the general solution for (-1/4)+cos^2(x)=1 ?

The general solution for (-1/4)+cos^2(x)=1 is No Solution for x\in\mathbb{R}

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Popular >

(-1/4)+cos^2(x)=1

Solution

(- \frac{1}{4}) + cos^{2} (x) = 1

Solution

No Solution for x \in R

Solution steps

(- \frac{1}{4}) + cos^{2} (x) = 1

Solve by substitution

- \frac{1}{4} + cos^{2} (x) = 1

Let:

cos (x) = u

- \frac{1}{4} + u^{2} = 1

- \frac{1}{4} + u^{2} = 1 : u = \frac{5}{2}, u = - \frac{5}{2}

- \frac{1}{4} + u^{2} = 1

Move

\frac{1}{4}

to the right side

- \frac{1}{4} + u^{2} = 1

Add

\frac{1}{4}

to both sides

- \frac{1}{4} + u^{2} + \frac{1}{4} = 1 + \frac{1}{4}

Simplify

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

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

Simplify

1 + \frac{1}{4} : \frac{5}{4}

1 + \frac{1}{4}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 4 + 1 = 5

1 \cdot 4 + 1

Multiply the numbers:

1 \cdot 4 = 4

= 4 + 1

Add the numbers:

4 + 1 = 5

= 5

= \frac{5}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{5}{4}, u = - \frac{5}{4}

\frac{5}{4} = \frac{5}{2}

\frac{5}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{5}{2}

- \frac{5}{4} = - \frac{5}{2}

- \frac{5}{4}

Simplify

\frac{5}{4} : \frac{5}{2}

\frac{5}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{5}{2}

= - \frac{5}{2}

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

Substitute back

u = cos (x)

cos (x) = \frac{5}{2}, cos (x) = - \frac{5}{2}

cos (x) = \frac{5}{2}, cos (x) = - \frac{5}{2}

cos (x) = \frac{5}{2} :

No Solution

cos (x) = \frac{5}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = - \frac{5}{2} :

No Solution

cos (x) = - \frac{5}{2}

- 1 \leq cos (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 (-1/4)+cos^2(x)=1 ?
The general solution for (-1/4)+cos^2(x)=1 is No Solution for x\in\mathbb{R}