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Popular Trigonometry >

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

Solution

cos (x) + 3 sin (x) \geq 2

Solution

\frac{π}{12} + 2 πn \leq x \leq \frac{7 π}{12} + 2 πn

Interval Notation

[\frac{π}{12} + 2 πn, \frac{7 π}{12} + 2 πn]

Decimal

0.26179 \dots + 2 πn \leq x \leq 1.83259 \dots + 2 πn

Solution steps

cos (x) + 3 sin (x) \geq 2

Rewrite using trig identities

Divide both sides by

2

\frac{cos ( x ) + 3 sin ( x )}{2} \geq \frac{2}{2}

Expand

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

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

Apply the fraction rule:

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

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

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

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

\frac{1}{2} cos (x) + \frac{3}{2} sin (x) \geq \frac{2}{2}

\frac{3}{2} = cos (\frac{π}{6})

\frac{1}{2} cos (x) + cos (\frac{π}{6}) sin (x) \geq \frac{2}{2}

\frac{1}{2} = sin (\frac{π}{6})

sin (\frac{π}{6}) cos (x) + cos (\frac{π}{6}) sin (x) \geq \frac{2}{2}

Use the following identity:

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

sin (\frac{π}{6} + x) \geq \frac{2}{2}

sin (\frac{π}{6} + x) \geq \frac{2}{2}

For

sin (x) \geq a

, if

- 1 < a < 1

then

arcsin (a) + 2 πn \leq x \leq π - arcsin (a) + 2 πn

arcsin (\frac{2}{2}) + 2 πn \leq (\frac{π}{6} + x) \leq π - arcsin (\frac{2}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

arcsin (\frac{2}{2}) + 2 πn \leq \frac{π}{6} + x and \frac{π}{6} + x \leq π - arcsin (\frac{2}{2}) + 2 πn

arcsin (\frac{2}{2}) + 2 πn \leq \frac{π}{6} + x : x \geq 2 πn + \frac{π}{12}

arcsin (\frac{2}{2}) + 2 πn \leq \frac{π}{6} + x

Switch sides

\frac{π}{6} + x \geq 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}{6} + x \geq \frac{π}{4} + 2 πn

Move

\frac{π}{6}

to the right side

\frac{π}{6} + x \geq \frac{π}{4} + 2 πn

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

\frac{π}{6} + x - \frac{π}{6} : x

\frac{π}{6} + x - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \geq 0

= x

Simplify

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

\frac{π}{4} + 2 πn - \frac{π}{6}

Group like terms

= 2 πn + \frac{π}{4} - \frac{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{π}{6} :

multiply the denominator and numerator by

2

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

= \frac{π 3}{12} - \frac{π 2}{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 2}{12}

Add similar elements:

3 π - 2 π = π

= 2 πn + \frac{π}{12}

x \geq 2 πn + \frac{π}{12}

x \geq 2 πn + \frac{π}{12}

x \geq 2 πn + \frac{π}{12}

\frac{π}{6} + x \leq π - arcsin (\frac{2}{2}) + 2 πn : x \leq \frac{7 π}{12} + 2 πn

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

Move

\frac{π}{6}

to the right side

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

Subtract

\frac{π}{6}

from both sides

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

Simplify

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

Simplify

\frac{π}{6} + x - \frac{π}{6} : x

\frac{π}{6} + x - \frac{π}{6}

Add similar elements:

\frac{π}{6} - \frac{π}{6} \leq 0

= x

Simplify

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

π - \frac{π}{4} + 2 πn - \frac{π}{6}

Group like terms

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

multiply the denominator and numerator by

2

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

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

Add similar elements:

- 3 π - 2 π = - 5 π

= \frac{- 5 π}{12}

Apply the fraction rule:

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

= π + 2 πn - \frac{5 π}{12}

x \leq π + 2 πn - \frac{5 π}{12}

x \leq π + 2 πn - \frac{5 π}{12}

x \leq π + 2 πn - \frac{5 π}{12}

Simplify

π - \frac{5 π}{12} : \frac{7 π}{12}

π - \frac{5 π}{12}

Convert element to fraction:

π = \frac{π 12}{12}

= \frac{π 12}{12} - \frac{5 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 - 5 π}{12}

Add similar elements:

12 π - 5 π = 7 π

= \frac{7 π}{12}

x \leq \frac{7 π}{12} + 2 πn

Combine the intervals

x \geq 2 πn + \frac{π}{12} and x \leq \frac{7 π}{12} + 2 πn

Merge Overlapping Intervals

\frac{π}{12} + 2 πn \leq x \leq \frac{7 π}{12} + 2 πn

Popular Examples

sec^2(x)<= tan^2(x)+sec(x)