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Popular >

tan(θ)=-4/3 \land sin(θ)<0,sec(θ)

Solution

tan (θ) = - \frac{4}{3} and sin (θ) < 0, sec (θ)

False for all θ \in R

Solution steps

tan (θ) = - \frac{4}{3} and sin (θ) < 0

tan (θ) = - \frac{4}{3} : θ = - 0.92729 \dots + πn

tan (θ) = - \frac{4}{3}

Apply trig inverse properties

tan (θ) = - \frac{4}{3}

General solutions for

tan (θ) = - \frac{4}{3}

tan (x) = - a \Rightarrow x = arctan (- a) + πn

θ = arctan (- \frac{4}{3}) + πn

θ = arctan (- \frac{4}{3}) + πn

Show solutions in decimal form

θ = - 0.92729 \dots + πn

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

Combine the intervals

θ = - 0.92729 \dots + πn and - π + 2 πn < θ < 2 πn

Merge Overlapping Intervals

False for all θ \in R

tan (x) 0 \leq x \leq \frac{π}{6}

cot (t) < 0 and sec (t) > 0

4 cos (θ) 4 sin (θ) 0 \leq θ \leq \frac{π}{2}

- 1 \geq cos (2 x) \geq 1

- \frac{1}{2} \leq sin (x) \leq \frac{3}{2}