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Popular Trigonometry >

cos(5x)cos(x/4)-sin(5x)sin(x/4)>=-(sqrt(2))/2

Solution

cos (5 x) cos (\frac{x}{4}) - sin (5 x) sin (\frac{x}{4}) \geq - \frac{2}{2}

Solution

- \frac{π}{7} + \frac{8 π}{21} n \leq x \leq \frac{π}{7} + \frac{8 π}{21} n

Interval Notation

[- \frac{π}{7} + \frac{8 π}{21} n, \frac{π}{7} + \frac{8 π}{21} n]

Decimal

- 0.44879 \dots + \frac{8 π}{21} n \leq x \leq 0.44879 \dots + \frac{8 π}{21} n

Solution steps

cos (5 x) cos (\frac{x}{4}) - sin (5 x) sin (\frac{x}{4}) \geq - \frac{2}{2}

Use the following identity:

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

cos (5 x + \frac{x}{4}) \geq - \frac{2}{2}

For

cos (x) \geq a

, if

- 1 < a < 1

then

- arccos (a) + 2 πn \leq x \leq arccos (a) + 2 πn

- arccos (- \frac{2}{2}) + 2 πn \leq (5 x + \frac{x}{4}) \leq arccos (- \frac{2}{2}) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

- arccos (- \frac{2}{2}) + 2 πn \leq 5 x + \frac{x}{4} and 5 x + \frac{x}{4} \leq arccos (- \frac{2}{2}) + 2 πn

- arccos (- \frac{2}{2}) + 2 πn \leq 5 x + \frac{x}{4} : x \geq - \frac{π}{7} + \frac{8 πn}{21}

- arccos (- \frac{2}{2}) + 2 πn \leq 5 x + \frac{x}{4}

Switch sides

5 x + \frac{x}{4} \geq - arccos (- \frac{2}{2}) + 2 πn

Simplify

- arccos (- \frac{2}{2}) + 2 πn : - \frac{3 π}{4} + 2 πn

- arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= - \frac{3 π}{4} + 2 πn

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

Multiply both sides by

4

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

Multiply both sides by

4

5 x \cdot 4 + \frac{x}{4} \cdot 4 \geq - \frac{3 π}{4} \cdot 4 + 2 πn \cdot 4

Simplify

5 x \cdot 4 + \frac{x}{4} \cdot 4 \geq - \frac{3 π}{4} \cdot 4 + 2 πn \cdot 4

Simplify

5 x \cdot 4 : 20 x

5 x \cdot 4

Multiply the numbers:

5 \cdot 4 = 20

= 20 x

Simplify

\frac{x}{4} \cdot 4 : x

\frac{x}{4} \cdot 4

Multiply fractions:

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

= \frac{x \cdot 4}{4}

Cancel the common factor:

4

= x

Simplify

- \frac{3 π}{4} \cdot 4 : - 3 π

- \frac{3 π}{4} \cdot 4

Multiply fractions:

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

= - \frac{3 π 4}{4}

Cancel the common factor:

4

= - 3 π

Simplify

2 πn \cdot 4 : 8 πn

2 πn \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 πn

20 x + x \geq - 3 π + 8 πn

21 x \geq - 3 π + 8 πn

21 x \geq - 3 π + 8 πn

21 x \geq - 3 π + 8 πn

Divide both sides by

21

21 x \geq - 3 π + 8 πn

Divide both sides by

21

\frac{21 x}{21} \geq - \frac{3 π}{21} + \frac{8 πn}{21}

Simplify

x \geq - \frac{π}{7} + \frac{8 πn}{21}

x \geq - \frac{π}{7} + \frac{8 πn}{21}

5 x + \frac{x}{4} \leq arccos (- \frac{2}{2}) + 2 πn : x \leq \frac{π}{7} + \frac{8 πn}{21}

5 x + \frac{x}{4} \leq arccos (- \frac{2}{2}) + 2 πn

Simplify

arccos (- \frac{2}{2}) + 2 πn : \frac{3 π}{4} + 2 πn

arccos (- \frac{2}{2}) + 2 πn

Use the following trivial identity:

arccos (- \frac{2}{2}) = \frac{3 π}{4}

x - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arccos (x) π \frac{5 π}{6} \frac{3 π}{4} \frac{2 π}{3} \frac{π}{2} \frac{π}{3} \frac{π}{4} \frac{π}{6} 0 arccos (x) 18 0^{\circ} 15 0^{\circ} 13 5^{\circ} 12 0^{\circ} 9 0^{\circ} 6 0^{\circ} 4 5^{\circ} 3 0^{\circ} 0^{\circ}

= \frac{3 π}{4} + 2 πn

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

Multiply both sides by

4

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

Multiply both sides by

4

5 x \cdot 4 + \frac{x}{4} \cdot 4 \leq \frac{3 π}{4} \cdot 4 + 2 πn \cdot 4

Simplify

5 x \cdot 4 + \frac{x}{4} \cdot 4 \leq \frac{3 π}{4} \cdot 4 + 2 πn \cdot 4

Simplify

5 x \cdot 4 : 20 x

5 x \cdot 4

Multiply the numbers:

5 \cdot 4 = 20

= 20 x

Simplify

\frac{x}{4} \cdot 4 : x

\frac{x}{4} \cdot 4

Multiply fractions:

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

= \frac{x \cdot 4}{4}

Cancel the common factor:

4

= x

Simplify

\frac{3 π}{4} \cdot 4 : 3 π

\frac{3 π}{4} \cdot 4

Multiply fractions:

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

= \frac{3 π 4}{4}

Cancel the common factor:

4

= 3 π

Simplify

2 πn \cdot 4 : 8 πn

2 πn \cdot 4

Multiply the numbers:

2 \cdot 4 = 8

= 8 πn

20 x + x \leq 3 π + 8 πn

21 x \leq 3 π + 8 πn

21 x \leq 3 π + 8 πn

21 x \leq 3 π + 8 πn

Divide both sides by

21

21 x \leq 3 π + 8 πn

Divide both sides by

21

\frac{21 x}{21} \leq \frac{3 π}{21} + \frac{8 πn}{21}

Simplify

x \leq \frac{π}{7} + \frac{8 πn}{21}

x \leq \frac{π}{7} + \frac{8 πn}{21}

Combine the intervals

x \geq - \frac{π}{7} + \frac{8 πn}{21} and x \leq \frac{π}{7} + \frac{8 πn}{21}

Merge Overlapping Intervals

- \frac{π}{7} + \frac{8 π}{21} n \leq x \leq \frac{π}{7} + \frac{8 π}{21} n

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