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Popular >

sin(θ)= 2/3 \land tan(θ)>0

Solution

sin (θ) = \frac{2}{3} and tan (θ) > 0

Solution

θ = 0.72972 \dots + 2 πn

+1

Interval Notation

θ = 0.72972 \dots + 2 πn

Solution steps

sin (θ) = \frac{2}{3} and tan (θ) > 0

sin (θ) = \frac{2}{3} : θ = 0.72972 \dots + 2 πn, θ = π - 0.72972 \dots + 2 πn

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

If

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

Merge Overlapping Intervals

θ = 0.72972 \dots + 2 πn

Popular Examples

cos(θ)<0\land tan(θ)>0

cos (θ) < 0 and tan (θ) > 0

cosh(θ)= 29/8 \land θ<0,sinh(θ)

cosh (θ) = \frac{29}{8} and θ < 0, sinh (θ)

0 < arcsin (y) < π

-1<= cos(x)<= 1

- 1 \leq cos (x) \leq 1

-pi/2 <arctan(y)<0

- \frac{π}{2} < arctan (y) < 0