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Popular Trigonometry >

tan(x)<0<5sin(x)

Solution

tan (x) < 0 < 5 sin (x)

Solution

- \frac{π}{2} + πn < x < πn

+2

Interval Notation

(- \frac{π}{2} + πn, πn)

Decimal

- 1.57079 \dots + πn < x < πn

Solution steps

tan (x) < 0 < 5 sin (x)

If

a < u < b

then

a < u and u < b

tan (x) < 0 and 0 < 5 sin (x)

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

tan (x) < 0

If

tan (x) < a

then

- \frac{π}{2} + πn < x < arctan (a) + πn

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

Simplify

arctan (0) : 0

arctan (0)

Use the following trivial identity:

arctan (0) = 0

x 0 \frac{3}{3} 13 arctan (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} arctan (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ}

= 0

- \frac{π}{2} + πn < x < 0 + πn

Simplify

- \frac{π}{2} + πn < x < πn

0 < 5 sin (x) : 2 πn < x < π + 2 πn

0 < 5 sin (x)

Switch sides

5 sin (x) > 0

Divide both sides by

5

5 sin (x) > 0

Divide both sides by

5

\frac{5 sin ( x )}{5} > \frac{0}{5}

Simplify

sin (x) > 0

sin (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 < x < π - arcsin (0) + 2 πn

Simplify

arcsin (0) : 0

arcsin (0)

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

Simplify

π - arcsin (0) : π

π - arcsin (0)

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

π - 0 = π

= π

0 + 2 πn < x < π + 2 πn

Simplify

2 πn < x < π + 2 πn

Combine the intervals

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

Merge Overlapping Intervals

- \frac{π}{2} + πn < x < πn

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