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Popular >

sin(θ)=-6/9 \land tan(θ)>0

Solution

sin (θ) = - \frac{6}{9} and tan (θ) > 0

False for all θ \in R

Solution steps

sin (θ) = - \frac{6}{9} and tan (θ) > 0

sin (θ) = - \frac{6}{9} : θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

sin (θ) = - \frac{6}{9}

Cancel the common factor:

3

sin (θ) = - \frac{2}{3}

Apply trig inverse properties

sin (θ) = - \frac{2}{3}

General solutions for

sin (θ) = - \frac{2}{3}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

θ = arcsin (- \frac{2}{3}) + 2 πn, θ = π + arcsin (\frac{2}{3}) + 2 πn

θ = arcsin (- \frac{2}{3}) + 2 πn, θ = π + arcsin (\frac{2}{3}) + 2 πn

Show solutions in decimal form

θ = - 0.72972 \dots + 2 πn, θ = π + 0.72972 \dots + 2 πn

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

tan (θ) > 0

tan (x) > a

then

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

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

Simplify

πn < θ < \frac{π}{2} + πn

Combine the intervals

(θ = π + 0.72972 \dots + 2 πn or θ = - 0.72972 \dots + 2 πn) and πn < θ < \frac{π}{2} + πn

Merge Overlapping Intervals

False for all θ \in R

cos (θ) = \frac{5}{7} and cot (θ) < 0, sin (θ)

sec (θ) < 0 and (cos (θ)) (sin (θ)) < 0

0 \leq a + b arctan (x) \leq 1

cos (x) < sin (x) < 1

cos (θ) = 45 and 0^{\circ} < θ < 9 0^{\circ}, sin (θ)