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Popular Trigonometry >

sin(180-x)<0

Solution

sin (18 0^{\circ} - x) < 0

Solution

18 0^{\circ} + 2 πn < x < π + 18 0^{\circ} + 2 πn

Interval Notation

(18 0^{\circ} + 2 πn, π + 18 0^{\circ} + 2 πn)

Decimal

3.14159 \dots + 2 πn < x < 6.28318 \dots + 2 πn

Solution steps

sin (18 0^{\circ} - x) < 0

Factor out

- 1

from

18 0^{\circ} - x : - (- 18 0^{\circ} + x)

sin (- (- 18 0^{\circ} + x)) < 0

Use the following identity:

sin (- x) = - sin (x)

- sin (- 18 0^{\circ} + x) < 0

Multiply both sides by

- 1

- sin (- 18 0^{\circ} + x) < 0

Multiply both sides by -1 (reverse the inequality)

(- sin (- 18 0^{\circ} + x)) (- 1) > 0 \cdot (- 1)

Simplify

sin (- 18 0^{\circ} + x) > 0

sin (- 18 0^{\circ} + x) > 0

For

sin (x) > a

, if

- 1 \leq a < 1

then

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

arcsin (0) + 2 πn < (- 18 0^{\circ} + x) < π - arcsin (0) + 2 πn

a < u < b

then

a < u and u < b

arcsin (0) + 2 πn < - 18 0^{\circ} + x and - 18 0^{\circ} + x < π - arcsin (0) + 2 πn

arcsin (0) + 2 πn < - 18 0^{\circ} + x : x > 2 πn + 18 0^{\circ}

arcsin (0) + 2 πn < - 18 0^{\circ} + x

Switch sides

- 18 0^{\circ} + 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

- 18 0^{\circ} + x > 2 πn

Move

18 0^{\circ}

to the right side

- 18 0^{\circ} + x > 2 πn

Add

18 0^{\circ}

to both sides

- 18 0^{\circ} + x + 18 0^{\circ} > 2 πn + 18 0^{\circ}

Simplify

x > 2 πn + 18 0^{\circ}

x > 2 πn + 18 0^{\circ}

- 18 0^{\circ} + x < π - arcsin (0) + 2 πn : x < π + 2 πn + 18 0^{\circ}

- 18 0^{\circ} + x < π - arcsin (0) + 2 πn

Simplify

π - arcsin (0) + 2 πn : π + 2 πn

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

= π + 2 πn

- 18 0^{\circ} + x < π + 2 πn

Move

18 0^{\circ}

to the right side

- 18 0^{\circ} + x < π + 2 πn

Add

18 0^{\circ}

to both sides

- 18 0^{\circ} + x + 18 0^{\circ} < π + 2 πn + 18 0^{\circ}

Simplify

x < π + 2 πn + 18 0^{\circ}

x < π + 2 πn + 18 0^{\circ}

Combine the intervals

x > 2 πn + 18 0^{\circ} and x < π + 2 πn + 18 0^{\circ}

Merge Overlapping Intervals

18 0^{\circ} + 2 πn < x < π + 18 0^{\circ} + 2 πn

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