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sec(θ)= 13/5 \land sin(θ)<0

Solution

sec (θ) = \frac{13}{5} and sin (θ) < 0

θ = 5.10718 \dots + 2 πn

Interval Notation

θ = 5.10718 \dots + 2 πn

Solution steps

sec (θ) = \frac{13}{5} and sin (θ) < 0

sec (θ) = \frac{13}{5} : θ = 1.17600 \dots + 2 πn, θ = 2 π - 1.17600 \dots + 2 πn

sec (θ) = \frac{13}{5}

Apply trig inverse properties

sec (θ) = \frac{13}{5}

General solutions for

sec (θ) = \frac{13}{5}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

θ = arcsec (\frac{13}{5}) + 2 πn, θ = 2 π - arcsec (\frac{13}{5}) + 2 πn

θ = arcsec (\frac{13}{5}) + 2 πn, θ = 2 π - arcsec (\frac{13}{5}) + 2 πn

Show solutions in decimal form

θ = 1.17600 \dots + 2 πn, θ = 2 π - 1.17600 \dots + 2 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 = 1.17600 \dots + 2 πn or θ = 2 π - 1.17600 \dots + 2 πn) and - π + 2 πn < θ < 2 πn

Merge Overlapping Intervals

θ = 5.10718 \dots + 2 πn

cot (θ) = \frac{2}{3} and csc (θ) < 0

- \frac{3}{2} < cos (x) < \frac{3}{2}

0 \leq - cos (b) 0.6428 \leq 180

- 3 < tan (θ) < 1

- \frac{1}{2} < cos (x) \leq \frac{1}{2}