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csc(x)=(-sqrt(13))/2 \land tan(x)>0

Solution

csc (x) = \frac{- 13}{2} and tan (x) > 0

False for all x \in R

Solution steps

csc (x) = \frac{- 13}{2} and tan (x) > 0

csc (x) = \frac{- 13}{2} : x = - 0.58800 \dots + 2 πn, x = π + 0.58800 \dots + 2 πn

csc (x) = \frac{- 13}{2}

Apply the fraction rule:

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

csc (x) = - \frac{13}{2}

Apply trig inverse properties

csc (x) = - \frac{13}{2}

General solutions for

csc (x) = - \frac{13}{2}

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- \frac{13}{2}) + 2 πn, x = π + arccsc (\frac{13}{2}) + 2 πn

x = arccsc (- \frac{13}{2}) + 2 πn, x = π + arccsc (\frac{13}{2}) + 2 πn

Show solutions in decimal form

x = - 0.58800 \dots + 2 πn, x = π + 0.58800 \dots + 2 πn

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

tan (x) > 0

tan (x) > a

then

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

arctan (0) + πn < x < \frac{π}{2} + 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

0 + πn < x < \frac{π}{2} + πn

Simplify

πn < x < \frac{π}{2} + πn

Combine the intervals

(x = π + 0.58800 \dots + 2 πn or x = - 0.58800 \dots + 2 πn) and πn < x < \frac{π}{2} + πn

Merge Overlapping Intervals

False for all x \in R

- 2 \leq 2 cos (3 x + 5) \leq 2

arccos (- 0.83) 180 < θ < 270

- \frac{1}{2} < sin (x) < 1

0 \leq x \leq 2 π arccos (\frac{1}{2})

24 sin (θ) > 0.06 > 24 cos (θ)