English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

cos(pi/4 x)>0

Solution

cos (\frac{π}{4} x) > 0

Solution

- 2 + 8 n < x < 2 + 8 n

Interval Notation

(- 2 + 8 n, 2 + 8 n)

Solution steps

cos (\frac{π}{4} x) > 0

For

cos (x) > a

, if

- 1 \leq a < 1

then

- arccos (a) + 2 πn < x < arccos (a) + 2 πn

- arccos (0) + 2 πn < \frac{π}{4} x < arccos (0) + 2 πn

a < u < b

then

a < u and u < b

- arccos (0) + 2 πn < \frac{π}{4} x and \frac{π}{4} x < arccos (0) + 2 πn

- arccos (0) + 2 πn < \frac{π}{4} x : x > - 2 + 8 n

- arccos (0) + 2 πn < \frac{π}{4} x

Switch sides

\frac{π}{4} x > - arccos (0) + 2 πn

Simplify

- arccos (0) + 2 πn : - \frac{π}{2} + 2 πn

- arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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{π}{2} + 2 πn

\frac{π}{4} x > - \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{π}{4} x > - \frac{π}{2} + 2 πn

Multiply both sides by

4

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

Simplify

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

Simplify

4 \cdot \frac{π}{4} x : π x

4 \cdot \frac{π}{4} x

Multiply fractions:

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

= \frac{4 π}{4} x

Cancel the common factor:

4

= x π

Simplify

- 4 \cdot \frac{π}{2} + 4 \cdot 2 πn : - 2 π + 8 πn

- 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= - 2 π + 8 πn

π x > - 2 π + 8 πn

π x > - 2 π + 8 πn

π x > - 2 π + 8 πn

Divide both sides by

π

π x > - 2 π + 8 πn

Divide both sides by

π

\frac{π x}{π} > - \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π x}{π} > - \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

- \frac{2 π}{π} + \frac{8 πn}{π} : - 2 + 8 n

- \frac{2 π}{π} + \frac{8 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= - 2 + \frac{8 πn}{π}

Cancel

\frac{8 πn}{π} : 8 n

\frac{8 πn}{π}

Cancel the common factor:

π

= 8 n

= - 2 + 8 n

x > - 2 + 8 n

x > - 2 + 8 n

x > - 2 + 8 n

\frac{π}{4} x < arccos (0) + 2 πn : x < 2 + 8 n

\frac{π}{4} x < arccos (0) + 2 πn

Simplify

arccos (0) + 2 πn : \frac{π}{2} + 2 πn

arccos (0) + 2 πn

Use the following trivial identity:

arccos (0) = \frac{π}{2}

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{π}{2} + 2 πn

\frac{π}{4} x < \frac{π}{2} + 2 πn

Multiply both sides by

4

\frac{π}{4} x < \frac{π}{2} + 2 πn

Multiply both sides by

4

4 \cdot \frac{π}{4} x < 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{π}{4} x < 4 \cdot \frac{π}{2} + 4 \cdot 2 πn

Simplify

4 \cdot \frac{π}{4} x : π x

4 \cdot \frac{π}{4} x

Multiply fractions:

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

= \frac{4 π}{4} x

Cancel the common factor:

4

= x π

Simplify

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

π x < 2 π + 8 πn

π x < 2 π + 8 πn

π x < 2 π + 8 πn

Divide both sides by

π

π x < 2 π + 8 πn

Divide both sides by

π

\frac{π x}{π} < \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π x}{π} < \frac{2 π}{π} + \frac{8 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{2 π}{π} + \frac{8 πn}{π} : 2 + 8 n

\frac{2 π}{π} + \frac{8 πn}{π}

Cancel

\frac{2 π}{π} : 2

\frac{2 π}{π}

Cancel the common factor:

π

= 2

= 2 + \frac{8 πn}{π}

Cancel

\frac{8 πn}{π} : 8 n

\frac{8 πn}{π}

Cancel the common factor:

π

= 8 n

= 2 + 8 n

x < 2 + 8 n

x < 2 + 8 n

x < 2 + 8 n

Combine the intervals

x > - 2 + 8 n and x < 2 + 8 n

Merge Overlapping Intervals

- 2 + 8 n < x < 2 + 8 n

Popular Examples

csc(-θ)<0

tan(x)<=-sqrt(3)

1-1/(cos(x))< 1/2*10^{-2}

sin(x)-sqrt(3)cos(x)<1

(cos(x)(1+tan(x)))/(cos(x)(1-tan(x)))>0