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Popular >

cos^2(x)=2cos^2(x/2)

Solution

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

Solution

x = 2 \cdot 1.11851 \dots + 4 πn, x = 4 π - 2 \cdot 1.11851 \dots + 4 πn, x = 2 \cdot 2.02307 \dots + 4 πn, x = - 2 \cdot 2.02307 \dots + 4 πn

Degrees

x = 128.17270 \dots^{\circ} + 72 0^{\circ} n, x = 591.82729 \dots^{\circ} + 72 0^{\circ} n, x = 231.82729 \dots^{\circ} + 72 0^{\circ} n, x = - 231.82729 \dots^{\circ} + 72 0^{\circ} n

Solution steps

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

Subtract

2 cos^{2} (\frac{x}{2})

from both sides

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

Let:

u = \frac{x}{2}

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

Factor

cos^{2} (2 u) - 2 cos^{2} (u) : (cos (2 u) + 2 cos (u)) (cos (2 u) - 2 cos (u))

cos^{2} (2 u) - 2 cos^{2} (u)

Rewrite

cos^{2} (2 u) - 2 cos^{2} (u)

cos^{2} (2 u) - (2 cos (u))^{2}

cos^{2} (2 u) - 2 cos^{2} (u)

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{2} cos^{2} (u) = (2 cos (u))^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

= (cos (2 u) + 2 cos (u)) (cos (2 u) - 2 cos (u))

(cos (2 u) + 2 cos (u)) (cos (2 u) - 2 cos (u)) = 0

Solving each part separately

cos (2 u) + 2 cos (u) = 0 or cos (2 u) - 2 cos (u) = 0

cos (2 u) + 2 cos (u) = 0 : u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

cos (2 u) + 2 cos (u) = 0

Rewrite using trig identities

cos (2 u) + cos (u) 2

Use the Double Angle identity:

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

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

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

Solve by substitution

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

Let:

cos (u) = u

- 1 + 2 u^{2} + u 2 = 0

- 1 + 2 u^{2} + u 2 = 0 : u = \frac{- 2 + 10}{4}, u = \frac{- 2 - 10}{4}

- 1 + 2 u^{2} + u 2 = 0

Write in the standard form

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

2 u^{2} + 2 u - 1 = 0

Solve with the quadratic formula

2 u^{2} + 2 u - 1 = 0

Quadratic Equation Formula:

For

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

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

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

(2)^{2} - 4 \cdot 2 (- 1) = 10

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

Apply rule

- (- a) = a

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

(2)^{2} = 2

(2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 8

= 2 + 8

Add the numbers:

2 + 8 = 10

= 10

u_{1, 2} = \frac{- 2 \pm 10}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 2 + 10}{2 \cdot 2}, u_{2} = \frac{- 2 - 10}{2 \cdot 2}

u = \frac{- 2 + 10}{2 \cdot 2} : \frac{- 2 + 10}{4}

\frac{- 2 + 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2 + 10}{4}

u = \frac{- 2 - 10}{2 \cdot 2} : \frac{- 2 - 10}{4}

\frac{- 2 - 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2 - 10}{4}

The solutions to the quadratic equation are:

u = \frac{- 2 + 10}{4}, u = \frac{- 2 - 10}{4}

Substitute back

u = cos (u)

cos (u) = \frac{- 2 + 10}{4}, cos (u) = \frac{- 2 - 10}{4}

cos (u) = \frac{- 2 + 10}{4}, cos (u) = \frac{- 2 - 10}{4}

cos (u) = \frac{- 2 + 10}{4} : u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

cos (u) = \frac{- 2 + 10}{4}

Apply trig inverse properties

cos (u) = \frac{- 2 + 10}{4}

General solutions for

cos (u) = \frac{- 2 + 10}{4}

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

u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

cos (u) = \frac{- 2 - 10}{4} :

No Solution

cos (u) = \frac{- 2 - 10}{4}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

cos (2 u) - 2 cos (u) = 0 : u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

cos (2 u) - 2 cos (u) = 0

Rewrite using trig identities

cos (2 u) - cos (u) 2

Use the Double Angle identity:

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

= 2 cos^{2} (u) - 1 - 2 cos (u)

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

Solve by substitution

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

Let:

cos (u) = u

- 1 + 2 u^{2} - u 2 = 0

- 1 + 2 u^{2} - u 2 = 0 : u = \frac{2 + 10}{4}, u = \frac{2 - 10}{4}

- 1 + 2 u^{2} - u 2 = 0

Write in the standard form

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

2 u^{2} - 2 u - 1 = 0

Solve with the quadratic formula

2 u^{2} - 2 u - 1 = 0

Quadratic Equation Formula:

For

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

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

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

(- 2)^{2} - 4 \cdot 2 (- 1) = 10

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

Apply rule

- (- a) = a

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

(- 2)^{2} = 2

(- 2)^{2}

Apply exponent rule:

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

n

is even

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

= (2)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 2

4 \cdot 2 \cdot 1 = 8

4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 8

= 2 + 8

Add the numbers:

2 + 8 = 10

= 10

u_{1, 2} = \frac{- ( - 2 ) \pm 10}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 10}{2 \cdot 2}, u_{2} = \frac{- ( - 2 ) - 10}{2 \cdot 2}

u = \frac{- ( - 2 ) + 10}{2 \cdot 2} : \frac{2 + 10}{4}

\frac{- ( - 2 ) + 10}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{2 + 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 + 10}{4}

u = \frac{- ( - 2 ) - 10}{2 \cdot 2} : \frac{2 - 10}{4}

\frac{- ( - 2 ) - 10}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{2 - 10}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 10}{4}

The solutions to the quadratic equation are:

u = \frac{2 + 10}{4}, u = \frac{2 - 10}{4}

Substitute back

u = cos (u)

cos (u) = \frac{2 + 10}{4}, cos (u) = \frac{2 - 10}{4}

cos (u) = \frac{2 + 10}{4}, cos (u) = \frac{2 - 10}{4}

cos (u) = \frac{2 + 10}{4} :

No Solution

cos (u) = \frac{2 + 10}{4}

- 1 \leq cos (x) \leq 1

No Solution

cos (u) = \frac{2 - 10}{4} : u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

cos (u) = \frac{2 - 10}{4}

Apply trig inverse properties

cos (u) = \frac{2 - 10}{4}

General solutions for

cos (u) = \frac{2 - 10}{4}

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

u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

Combine all the solutions

u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

Combine all the solutions

u = arccos (\frac{- 2 + 10}{4}) + 2 πn, u = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn, u = arccos (\frac{2 - 10}{4}) + 2 πn, u = - arccos (\frac{2 - 10}{4}) + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = arccos (\frac{- 2 + 10}{4}) + 2 πn : x = 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

\frac{x}{2} = arccos (\frac{- 2 + 10}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arccos (\frac{- 2 + 10}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn : 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccos (\frac{10 - 2}{4}) + 4 πn

x = 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

x = 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

x = 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn : x = 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = 2 π - arccos (\frac{- 2 + 10}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot 2 π - 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot 2 π - 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn : 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

2 \cdot 2 π - 2 arccos (\frac{- 2 + 10}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 π - 2 arccos (\frac{10 - 2}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

x = 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn

\frac{x}{2} = arccos (\frac{2 - 10}{4}) + 2 πn : x = 2 arccos (\frac{2 - 10}{4}) + 4 πn

\frac{x}{2} = arccos (\frac{2 - 10}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = arccos (\frac{2 - 10}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn : 2 arccos (\frac{2 - 10}{4}) + 4 πn

2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = 2 arccos (\frac{2 - 10}{4}) + 4 πn

\frac{x}{2} = - arccos (\frac{2 - 10}{4}) + 2 πn : x = - 2 arccos (\frac{2 - 10}{4}) + 4 πn

\frac{x}{2} = - arccos (\frac{2 - 10}{4}) + 2 πn

Multiply both sides by

2

\frac{x}{2} = - arccos (\frac{2 - 10}{4}) + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = - 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = - 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn : - 2 arccos (\frac{2 - 10}{4}) + 4 πn

- 2 arccos (\frac{2 - 10}{4}) + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= - 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = - 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = - 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = - 2 arccos (\frac{2 - 10}{4}) + 4 πn

x = 2 arccos (\frac{- 2 + 10}{4}) + 4 πn, x = 4 π - 2 arccos (\frac{- 2 + 10}{4}) + 4 πn, x = 2 arccos (\frac{2 - 10}{4}) + 4 πn, x = - 2 arccos (\frac{2 - 10}{4}) + 4 πn

Show solutions in decimal form

x = 2 \cdot 1.11851 \dots + 4 πn, x = 4 π - 2 \cdot 1.11851 \dots + 4 πn, x = 2 \cdot 2.02307 \dots + 4 πn, x = - 2 \cdot 2.02307 \dots + 4 πn

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