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Popular Trigonometry >

sqrt(3)cos(4x)+sin(4x)>sqrt(2)

Solution

3 cos (4 x) + sin (4 x) > 2

Solution

No Solution for x \in R

Solution steps

3 cos (4 x) + sin (4 x) > 2

Let:

u = 4 x

3 cos (u) + sin (u) > 2

3 cos (u) + sin (u) > 2 : - \frac{π}{12} + 2 πn < u < \frac{5 π}{12} + 2 πn

3 cos (u) + sin (u) > 2

Rewrite using trig identities

Divide both sides by

2

\frac{3 cos ( u ) + sin ( u )}{2} > \frac{2}{2}

Expand

\frac{3 cos ( u ) + sin ( u )}{2} : \frac{3}{2} cos (u) + \frac{1}{2} sin (u)

\frac{3 cos ( u ) + sin ( u )}{2}

Apply the fraction rule:

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

\frac{3 cos ( u ) + sin ( u )}{2} = \frac{3 cos ( u )}{2} + \frac{sin ( u )}{2}

= \frac{3 cos ( u )}{2} + \frac{sin ( u )}{2}

= \frac{3}{2} cos (u) + \frac{1}{2} sin (u)

\frac{3}{2} cos (u) + \frac{1}{2} sin (u) > \frac{2}{2}

\frac{3}{2} = sin (\frac{π}{3})

sin (\frac{π}{3}) cos (u) + \frac{1}{2} sin (u) > \frac{2}{2}

\frac{1}{2} = cos (\frac{π}{3})

sin (\frac{π}{3}) cos (u) + cos (\frac{π}{3}) sin (u) > \frac{2}{2}

Use the following identity:

cos (s) sin (t) + cos (t) sin (s) = sin (s + t)

sin (\frac{π}{3} + u) > \frac{2}{2}

sin (\frac{π}{3} + u) > \frac{2}{2}

For

sin (x) > a

, if

- 1 \leq a < 1

then

arcsin (a) + 2 πn < x < π - arcsin (a) + 2 πn

arcsin (\frac{2}{2}) + 2 πn < (\frac{π}{3} + u) < π - arcsin (\frac{2}{2}) + 2 πn

a < u < b

then

a < u and u < b

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{3} + u and \frac{π}{3} + u < π - arcsin (\frac{2}{2}) + 2 πn

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{3} + u : u > 2 πn - \frac{π}{12}

arcsin (\frac{2}{2}) + 2 πn < \frac{π}{3} + u

Switch sides

\frac{π}{3} + u > arcsin (\frac{2}{2}) + 2 πn

Simplify

arcsin (\frac{2}{2}) + 2 πn : \frac{π}{4} + 2 πn

arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4} + 2 πn

\frac{π}{3} + u > \frac{π}{4} + 2 πn

Move

\frac{π}{3}

to the right side

\frac{π}{3} + u > \frac{π}{4} + 2 πn

Subtract

\frac{π}{3}

from both sides

\frac{π}{3} + u - \frac{π}{3} > \frac{π}{4} + 2 πn - \frac{π}{3}

Simplify

\frac{π}{3} + u - \frac{π}{3} > \frac{π}{4} + 2 πn - \frac{π}{3}

Simplify

\frac{π}{3} + u - \frac{π}{3} : u

\frac{π}{3} + u - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} > 0

= u

Simplify

\frac{π}{4} + 2 πn - \frac{π}{3} : 2 πn - \frac{π}{12}

\frac{π}{4} + 2 πn - \frac{π}{3}

Group like terms

= 2 πn + \frac{π}{4} - \frac{π}{3}

Least Common Multiplier of

4, 3 : 12

4, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

4

3

= 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{π}{3} :

multiply the denominator and numerator by

4

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

= \frac{π 3}{12} - \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 - π 4}{12}

Add similar elements:

3 π - 4 π = - π

= \frac{- π}{12}

Apply the fraction rule:

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

= 2 πn - \frac{π}{12}

u > 2 πn - \frac{π}{12}

u > 2 πn - \frac{π}{12}

u > 2 πn - \frac{π}{12}

\frac{π}{3} + u < π - arcsin (\frac{2}{2}) + 2 πn : u < \frac{5 π}{12} + 2 πn

\frac{π}{3} + u < π - arcsin (\frac{2}{2}) + 2 πn

Simplify

π - arcsin (\frac{2}{2}) + 2 πn : π - \frac{π}{4} + 2 πn

π - arcsin (\frac{2}{2}) + 2 πn

Use the following trivial identity:

arcsin (\frac{2}{2}) = \frac{π}{4}

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= π - \frac{π}{4} + 2 πn

\frac{π}{3} + u < π - \frac{π}{4} + 2 πn

Move

\frac{π}{3}

to the right side

\frac{π}{3} + u < π - \frac{π}{4} + 2 πn

Subtract

\frac{π}{3}

from both sides

\frac{π}{3} + u - \frac{π}{3} < π - \frac{π}{4} + 2 πn - \frac{π}{3}

Simplify

\frac{π}{3} + u - \frac{π}{3} < π - \frac{π}{4} + 2 πn - \frac{π}{3}

Simplify

\frac{π}{3} + u - \frac{π}{3} : u

\frac{π}{3} + u - \frac{π}{3}

Add similar elements:

\frac{π}{3} - \frac{π}{3} < 0

= u

Simplify

π - \frac{π}{4} + 2 πn - \frac{π}{3} : π + 2 πn - \frac{7 π}{12}

π - \frac{π}{4} + 2 πn - \frac{π}{3}

Group like terms

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

Least Common Multiplier of

4, 3 : 12

4, 3

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

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

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

4

3

= 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{π}{3} :

multiply the denominator and numerator by

4

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

= - \frac{π 3}{12} - \frac{π 4}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 3 - π 4}{12}

Add similar elements:

- 3 π - 4 π = - 7 π

= \frac{- 7 π}{12}

Apply the fraction rule:

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

= π + 2 πn - \frac{7 π}{12}

u < π + 2 πn - \frac{7 π}{12}

u < π + 2 πn - \frac{7 π}{12}

u < π + 2 πn - \frac{7 π}{12}

Simplify

π - \frac{7 π}{12} : \frac{5 π}{12}

π - \frac{7 π}{12}

Convert element to fraction:

π = \frac{π 12}{12}

= \frac{π 12}{12} - \frac{7 π}{12}

Since the denominators are equal, combine the fractions:

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

= \frac{π 12 - 7 π}{12}

Add similar elements:

12 π - 7 π = 5 π

= \frac{5 π}{12}

u < \frac{5 π}{12} + 2 πn

Combine the intervals

u > 2 πn - \frac{π}{12} and u < \frac{5 π}{12} + 2 πn

Merge Overlapping Intervals

- \frac{π}{12} + 2 πn < u < \frac{5 π}{12} + 2 πn

- \frac{π}{12} + 2 πn < u < \frac{5 π}{12} + 2 πn

Substitute back

4 x = u

- \frac{π}{12} + 2 πn < 4 x < \frac{5 π}{12} + 2 πn

- \frac{π}{12} + 2 πn < 4 x < \frac{5 π}{12} + 2 πn :

False for all

x \in R

- \frac{π}{12} + 2 πn < 4 x < \frac{5 π}{12} + 2 πn

a < u < b

then

a < u and u < b

- \frac{π}{12} + 2 πn < 4 x and 4 x < \frac{5 π}{12} + 2 πn

- \frac{π}{12} + 2 πn < 4 x : x > - \frac{π}{48} + \frac{πn}{2}

- \frac{π}{12} + 2 πn < 4 x

Switch sides

4 x > - \frac{π}{12} + 2 πn

Divide both sides by

4

4 x > - \frac{π}{12} + 2 πn

Divide both sides by

4

\frac{4 x}{4} > - \frac{\frac{π}{12}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} > - \frac{\frac{π}{12}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

- \frac{\frac{π}{12}}{4} + \frac{2 πn}{4} : - \frac{π}{48} + \frac{πn}{2}

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

\frac{\frac{π}{12}}{4} = \frac{π}{48}

\frac{\frac{π}{12}}{4}

Apply the fraction rule:

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

= \frac{π}{12 \cdot 4}

Multiply the numbers:

12 \cdot 4 = 48

= \frac{π}{48}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= - \frac{π}{48} + \frac{πn}{2}

x > - \frac{π}{48} + \frac{πn}{2}

x > - \frac{π}{48} + \frac{πn}{2}

x > - \frac{π}{48} + \frac{πn}{2}

4 x < \frac{5 π}{12} + 2 πn : x < \frac{5 π}{48} + \frac{πn}{2}

4 x < \frac{5 π}{12} + 2 πn

Divide both sides by

4

4 x < \frac{5 π}{12} + 2 πn

Divide both sides by

4

\frac{4 x}{4} < \frac{\frac{5 π}{12}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} < \frac{\frac{5 π}{12}}{4} + \frac{2 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{\frac{5 π}{12}}{4} + \frac{2 πn}{4} : \frac{5 π}{48} + \frac{πn}{2}

\frac{\frac{5 π}{12}}{4} + \frac{2 πn}{4}

\frac{\frac{5 π}{12}}{4} = \frac{5 π}{48}

\frac{\frac{5 π}{12}}{4}

Apply the fraction rule:

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

= \frac{5 π}{12 \cdot 4}

Multiply the numbers:

12 \cdot 4 = 48

= \frac{5 π}{48}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

= \frac{5 π}{48} + \frac{πn}{2}

x < \frac{5 π}{48} + \frac{πn}{2}

x < \frac{5 π}{48} + \frac{πn}{2}

x < \frac{5 π}{48} + \frac{πn}{2}

Combine the intervals

x > - \frac{π}{48} + \frac{π}{2} n and x < \frac{5 π}{48} + \frac{π}{2} n

Merge Overlapping Intervals

False for all x \in R

No Solution for x \in R

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