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cos^2(3x+pi/4)= 3/4

Solution

cos^{2} (3 x + \frac{π}{4}) = \frac{3}{4}

Solution

x = \frac{2 πn}{3} - \frac{π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}, x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

Degrees

x = - 5^{\circ} + 12 0^{\circ} n, x = 9 5^{\circ} + 12 0^{\circ} n, x = 3 5^{\circ} + 12 0^{\circ} n, x = 5 5^{\circ} + 12 0^{\circ} n

Solution steps

cos^{2} (3 x + \frac{π}{4}) = \frac{3}{4}

Solve by substitution

cos^{2} (3 x + \frac{π}{4}) = \frac{3}{4}

Let:

cos (3 x + \frac{π}{4}) = u

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

Substitute back

u = cos (3 x + \frac{π}{4})

cos (3 x + \frac{π}{4}) = \frac{3}{2}, cos (3 x + \frac{π}{4}) = - \frac{3}{2}

cos (3 x + \frac{π}{4}) = \frac{3}{2}, cos (3 x + \frac{π}{4}) = - \frac{3}{2}

cos (3 x + \frac{π}{4}) = \frac{3}{2} : x = \frac{2 πn}{3} - \frac{π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}

cos (3 x + \frac{π}{4}) = \frac{3}{2}

General solutions for

cos (3 x + \frac{π}{4}) = \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

3 x + \frac{π}{4} = \frac{π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

3 x + \frac{π}{4} = \frac{π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Solve

3 x + \frac{π}{4} = \frac{π}{6} + 2 πn : x = \frac{2 πn}{3} - \frac{π}{36}

3 x + \frac{π}{4} = \frac{π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{π}{6} + 2 πn - \frac{π}{4} : 2 πn - \frac{π}{12}

\frac{π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn + \frac{π}{6} - \frac{π}{4}

Least Common Multiplier of

6, 4 : 12

6, 4

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Multiply each factor the greatest number of times it occurs in either

6

4

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{6} :

multiply the denominator and numerator by

2

\frac{π}{6} = \frac{π 2}{6 \cdot 2} = \frac{π 2}{12}

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

= \frac{π 2}{12} - \frac{π 3}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π 3}{12}

Add similar elements:

2 π - 3 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= 2 πn - \frac{π}{12}

3 x = 2 πn - \frac{π}{12}

3 x = 2 πn - \frac{π}{12}

3 x = 2 πn - \frac{π}{12}

Divide both sides by

3

3 x = 2 πn - \frac{π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} - \frac{\frac{π}{12}}{3} : \frac{2 πn}{3} - \frac{π}{36}

\frac{2 πn}{3} - \frac{\frac{π}{12}}{3}

\frac{\frac{π}{12}}{3} = \frac{π}{36}

\frac{\frac{π}{12}}{3}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{π}{36}

= \frac{2 πn}{3} - \frac{π}{36}

x = \frac{2 πn}{3} - \frac{π}{36}

x = \frac{2 πn}{3} - \frac{π}{36}

x = \frac{2 πn}{3} - \frac{π}{36}

Solve

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{19 π}{36}

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{11 π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{11 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{11 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{11 π}{6} + 2 πn - \frac{π}{4} : 2 πn + \frac{19 π}{12}

\frac{11 π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{11 π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{11 π}{6} :

multiply the denominator and numerator by

2

\frac{11 π}{6} = \frac{11 π 2}{6 \cdot 2} = \frac{22 π}{12}

= - \frac{π 3}{12} + \frac{22 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 22 π}{12}

Add similar elements:

- 3 π + 22 π = 19 π

= 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

3 x = 2 πn + \frac{19 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{19 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3} : \frac{2 πn}{3} + \frac{19 π}{36}

\frac{2 πn}{3} + \frac{\frac{19 π}{12}}{3}

\frac{\frac{19 π}{12}}{3} = \frac{19 π}{36}

\frac{\frac{19 π}{12}}{3}

Apply the fraction rule:

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

= \frac{19 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{19 π}{36}

= \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} + \frac{19 π}{36}

x = \frac{2 πn}{3} - \frac{π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}

cos (3 x + \frac{π}{4}) = - \frac{3}{2} : x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

cos (3 x + \frac{π}{4}) = - \frac{3}{2}

General solutions for

cos (3 x + \frac{π}{4}) = - \frac{3}{2}

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

3 x + \frac{π}{4} = \frac{5 π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

3 x + \frac{π}{4} = \frac{5 π}{6} + 2 πn, 3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

Solve

3 x + \frac{π}{4} = \frac{5 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{7 π}{36}

3 x + \frac{π}{4} = \frac{5 π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{5 π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{5 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{5 π}{6} + 2 πn - \frac{π}{4} : 2 πn + \frac{7 π}{12}

\frac{5 π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{5 π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{5 π}{6} :

multiply the denominator and numerator by

2

\frac{5 π}{6} = \frac{5 π 2}{6 \cdot 2} = \frac{10 π}{12}

= - \frac{π 3}{12} + \frac{10 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 10 π}{12}

Add similar elements:

- 3 π + 10 π = 7 π

= 2 πn + \frac{7 π}{12}

3 x = 2 πn + \frac{7 π}{12}

3 x = 2 πn + \frac{7 π}{12}

3 x = 2 πn + \frac{7 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{7 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3} : \frac{2 πn}{3} + \frac{7 π}{36}

\frac{2 πn}{3} + \frac{\frac{7 π}{12}}{3}

\frac{\frac{7 π}{12}}{3} = \frac{7 π}{36}

\frac{\frac{7 π}{12}}{3}

Apply the fraction rule:

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

= \frac{7 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{7 π}{36}

= \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}

Solve

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn : x = \frac{2 πn}{3} + \frac{11 π}{36}

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

Move

\frac{π}{4}

to the right side

3 x + \frac{π}{4} = \frac{7 π}{6} + 2 πn

Subtract

\frac{π}{4}

from both sides

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} = \frac{7 π}{6} + 2 πn - \frac{π}{4}

Simplify

3 x + \frac{π}{4} - \frac{π}{4} : 3 x

3 x + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= 3 x

Simplify

\frac{7 π}{6} + 2 πn - \frac{π}{4} : 2 πn + \frac{11 π}{12}

\frac{7 π}{6} + 2 πn - \frac{π}{4}

Group like terms

= 2 πn - \frac{π}{4} + \frac{7 π}{6}

Least Common Multiplier of

4, 6 : 12

4, 6

Least Common Multiplier (LCM)

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

4

6

= 2 \cdot 2 \cdot 3

Multiply the numbers:

2 \cdot 2 \cdot 3 = 12

= 12

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

12

For

\frac{π}{4} :

multiply the denominator and numerator by

3

\frac{π}{4} = \frac{π 3}{4 \cdot 3} = \frac{π 3}{12}

For

\frac{7 π}{6} :

multiply the denominator and numerator by

2

\frac{7 π}{6} = \frac{7 π 2}{6 \cdot 2} = \frac{14 π}{12}

= - \frac{π 3}{12} + \frac{14 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 + 14 π}{12}

Add similar elements:

- 3 π + 14 π = 11 π

= 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

3 x = 2 πn + \frac{11 π}{12}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3} : \frac{2 πn}{3} + \frac{11 π}{36}

\frac{2 πn}{3} + \frac{\frac{11 π}{12}}{3}

\frac{\frac{11 π}{12}}{3} = \frac{11 π}{36}

\frac{\frac{11 π}{12}}{3}

Apply the fraction rule:

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

= \frac{11 π}{12 \cdot 3}

Multiply the numbers:

12 \cdot 3 = 36

= \frac{11 π}{36}

= \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{11 π}{36}

x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

Combine all the solutions

x = \frac{2 πn}{3} - \frac{π}{36}, x = \frac{2 πn}{3} + \frac{19 π}{36}, x = \frac{2 πn}{3} + \frac{7 π}{36}, x = \frac{2 πn}{3} + \frac{11 π}{36}

Popular Examples

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arccos (e^{- x}) = \frac{π}{3}

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