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Popular >

sin(θ)<0\land sec(θ)>0

Solution

sin (θ) < 0 and sec (θ) > 0

Solution

False for all θ \in R

Solution steps

sin (θ) < 0 and sec (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

sec (θ) > 0 :

False for all

θ \in R

sec (θ) > 0

Express with sin, cos

sec (θ) > 0

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

\frac{1}{cos ( θ )} > 0

\frac{1}{cos ( θ )} > 0

If

\frac{1}{a} > 0

then

a > 0

cos (θ) > 0

No Solution for θ \in R

Combine the intervals

- π + 2 πn < θ < 2 πn and False for all θ \in R

Merge Overlapping Intervals

False for all θ \in R

Popular Examples

cos(θ)= 3/4 \land cot(θ)<0

cos (θ) = \frac{3}{4} and cot (θ) < 0

sin(θ)>0\land cos(θ)>0

sin (θ) > 0 and cos (θ) > 0

sin(4θ)0<= θ<= pi

sin (4 θ) 0 \leq θ \leq π

[sin(pi)t]0<= t<= 2

[sin (π) t] 0 \leq t \leq 2

cosh(θ)= 7/3 \land θ<0,sinh(θ)

cosh (θ) = \frac{7}{3} and θ < 0, sinh (θ)