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Popular Trigonometry >

-5(sin(x)+cos(x))>0

Solution

- 5 (sin (x) + cos (x)) > 0

Solution

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

Interval Notation

(- \frac{5 π}{4} + 2 πn, - \frac{π}{4} + 2 πn)

Decimal

- 3.92699 \dots + 2 πn < x < - 0.78539 \dots + 2 πn

Solution steps

- 5 (sin (x) + cos (x)) > 0

Use the following identity:

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

- 52 sin (\frac{π}{4} + x) > 0

Multiply both sides by

- 1

- 52 sin (\frac{π}{4} + x) > 0

Multiply both sides by -1 (reverse the inequality)

(- 52 sin (\frac{π}{4} + x)) (- 1) < 0 \cdot (- 1)

Simplify

52 sin (\frac{π}{4} + x) < 0

52 sin (\frac{π}{4} + x) < 0

Divide both sides by

52

52 sin (\frac{π}{4} + x) < 0

Divide both sides by

52

\frac{5 2 sin ( \frac{π}{4} + x )}{5 2} < \frac{0}{5 2}

Simplify

sin (\frac{π}{4} + x) < 0

sin (\frac{π}{4} + x) < 0

For

sin (x) < a

, if

- 1 < a \leq 1

then

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

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

a < u < b

then

a < u and u < b

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

- π - arcsin (0) + 2 πn < \frac{π}{4} + x : x > - \frac{5 π}{4} + 2 πn

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

Switch sides

\frac{π}{4} + x > - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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 - 0 + 2 πn

- π - 0 + 2 πn = - π + 2 πn

= 2 πn - π

\frac{π}{4} + x > 2 πn - π

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x > 2 πn - π

Subtract

\frac{π}{4}

from both sides

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

Simplify

x > 2 πn - π - \frac{π}{4}

x > 2 πn - π - \frac{π}{4}

Simplify

- π - \frac{π}{4} : - \frac{5 π}{4}

- π - \frac{π}{4}

Convert element to fraction:

π = \frac{π 4}{4}

= - \frac{π 4}{4} - \frac{π}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{- π 4 - π}{4}

Add similar elements:

- 4 π - π = - 5 π

= \frac{- 5 π}{4}

Apply the fraction rule:

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

= - \frac{5 π}{4}

x > - \frac{5 π}{4} + 2 πn

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

\frac{π}{4} + x < arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

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}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

\frac{π}{4} + x < 2 πn

Move

\frac{π}{4}

to the right side

\frac{π}{4} + x < 2 πn

Subtract

\frac{π}{4}

from both sides

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

Simplify

x < 2 πn - \frac{π}{4}

x < 2 πn - \frac{π}{4}

Combine the intervals

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

Merge Overlapping Intervals

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

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