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Popular >

sin(θ)<0\land cot(θ)<0

Solution

sin (θ) < 0 and cot (θ) < 0

Solution

False for all θ \in R

Solution steps

sin (θ) < 0 and cot (θ) < 0

sin (θ) < 0 : - π + 2 πn < θ < 2 πn

sin (θ) < 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 < θ < arcsin (0) + 2 πn

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

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

- π + 2 πn < θ < 0 + 2 πn

Simplify

- π + 2 πn < θ < 2 πn

cot (θ) < 0 : \frac{π}{2} + πn < θ < π + πn

cot (θ) < 0

If

cot (x) < a

then

arccot (a) + πn < x < π + πn

arccot (0) + πn < θ < π + πn

Simplify

arccot (0) : \frac{π}{2}

arccot (0)

Use the following trivial identity:

arccot (0) = \frac{π}{2}

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

= \frac{π}{2}

\frac{π}{2} + πn < θ < π + πn

Combine the intervals

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

Merge Overlapping Intervals

False for all θ \in R

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