What is the general solution for (sin^2(x)-2cos(x)+1)/4 =0 ?

The general solution for (sin^2(x)-2cos(x)+1)/4 =0 is x=0.74946…+2pin,x=2pi-0.74946…+2pin

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(sin^2(x)-2cos(x)+1)/4 =0

Solution

\frac{sin ^{2} ( x ) - 2 cos ( x ) + 1}{4} = 0

Solution

x = 0.74946 \dots + 2 πn, x = 2 π - 0.74946 \dots + 2 πn

Degrees

x = 42.94140 \dots^{\circ} + 36 0^{\circ} n, x = 317.05859 \dots^{\circ} + 36 0^{\circ} n

Solution steps

\frac{sin ^{2} ( x ) - 2 cos ( x ) + 1}{4} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin^{2} (x) - 2 cos (x) + 1 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Solve by substitution

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

Let:

cos (x) = u

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

2 - u^{2} - 2 u = 0 : u = - 1 - 3, u = 3 - 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 1, b = - 2, c = 2

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

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

(- 2)^{2} - 4 (- 1) \cdot 2 = 23

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 2^{2} + 8

2^{2} = 4

= 4 + 8

Add the numbers:

4 + 8 = 12

= 12

Prime factorization of

12 : 2^{2} \cdot 3

12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

Apply radical rule:

= 3 2^{2}

Apply radical rule:

2^{2} = 2

= 23

u_{1, 2} = \frac{- ( - 2 ) \pm 2 3}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 3}{2 ( - 1 )}, u_{2} = \frac{- ( - 2 ) - 2 3}{2 ( - 1 )}

u = \frac{- ( - 2 ) + 2 3}{2 ( - 1 )} : - 1 - 3

\frac{- ( - 2 ) + 2 3}{2 ( - 1 )}

Remove parentheses:

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

= \frac{2 + 2 3}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

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

Cancel

\frac{2 + 2 3}{2} : 1 + 3

\frac{2 + 2 3}{2}

Factor

2 + 23 : 2 (1 + 3)

2 + 23

Rewrite as

= 2 \cdot 1 + 23

Factor out common term

2

= 2 (1 + 3)

= \frac{2 ( 1 + 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 1 + 3

= - (1 + 3)

Distribute parentheses

= - (1) - (3)

Apply minus-plus rules

+ (- a) = - a

= - 1 - 3

u = \frac{- ( - 2 ) - 2 3}{2 ( - 1 )} : 3 - 1

\frac{- ( - 2 ) - 2 3}{2 ( - 1 )}

Remove parentheses:

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

Apply the fraction rule:

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

2 - 23 = - (23 - 2)

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

Factor

23 - 2 : 2 (3 - 1)

23 - 2

Rewrite as

= 23 - 2 \cdot 1

Factor out common term

2

= 2 (3 - 1)

= \frac{2 ( 3 - 1 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= 3 - 1

The solutions to the quadratic equation are:

u = - 1 - 3, u = 3 - 1

Substitute back

u = cos (x)

cos (x) = - 1 - 3, cos (x) = 3 - 1

cos (x) = - 1 - 3, cos (x) = 3 - 1

cos (x) = - 1 - 3 :

No Solution

cos (x) = - 1 - 3

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = 3 - 1 : x = arccos (3 - 1) + 2 πn, x = 2 π - arccos (3 - 1) + 2 πn

cos (x) = 3 - 1

Apply trig inverse properties

cos (x) = 3 - 1

General solutions for

cos (x) = 3 - 1

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (3 - 1) + 2 πn, x = 2 π - arccos (3 - 1) + 2 πn

x = arccos (3 - 1) + 2 πn, x = 2 π - arccos (3 - 1) + 2 πn

Combine all the solutions

x = arccos (3 - 1) + 2 πn, x = 2 π - arccos (3 - 1) + 2 πn

Show solutions in decimal form

x = 0.74946 \dots + 2 πn, x = 2 π - 0.74946 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for (sin^2(x)-2cos(x)+1)/4 =0 ?
The general solution for (sin^2(x)-2cos(x)+1)/4 =0 is x=0.74946…+2pin,x=2pi-0.74946…+2pin