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Popular >

sec(θ)<0\land (cos(θ))(sin(θ))<0

Solution

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

Solution

False for all θ \in R

Solution steps

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

sec (θ) < 0 :

False for all

θ \in R

sec (θ) < 0

Express with sin, cos

sec (θ) < 0

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

\frac{1}{cos ( θ )} < 0

\frac{1}{cos ( θ )} < 0

If

\frac{1}{a} < 0

then

a < 0

cos (θ) < 0

No Solution for θ \in R

cos (θ) sin (θ) < 0 : - \frac{π}{2} + πn < θ < πn

cos (θ) sin (θ) < 0

Use the following identity:

2 cos (x) sin (x) = sin (2 x)

Therefore

cos (x) sin (x) = \frac{sin ( 2 x )}{2}

\frac{sin ( 2 θ )}{2} < 0

Multiply both sides by

2

\frac{sin ( 2 θ )}{2} < 0

Multiply both sides by

2

\frac{2 sin ( 2 θ )}{2} < 0 \cdot 2

Simplify

sin (2 θ) < 0

sin (2 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 < 2 θ < arcsin (0) + 2 πn

If

a < u < b

then

a < u and u < b

- π - arcsin (0) + 2 πn < 2 θ and 2 θ < arcsin (0) + 2 πn

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

- π - arcsin (0) + 2 πn < 2 θ

Switch sides

2 θ > - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- π - arcsin (0) + 2 πn

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 + 2 πn

- π - 0 + 2 πn = - π + 2 πn

= 2 πn - π

2 θ > 2 πn - π

Divide both sides by

2

2 θ > 2 πn - π

Divide both sides by

2

\frac{2 θ}{2} > \frac{2 πn}{2} - \frac{π}{2}

Simplify

θ > πn - \frac{π}{2}

θ > πn - \frac{π}{2}

2 θ < arcsin (0) + 2 πn : θ < πn

2 θ < arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

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 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

2 θ < 2 πn

Divide both sides by

2

2 θ < 2 πn

Divide both sides by

2

\frac{2 θ}{2} < \frac{2 πn}{2}

Simplify

θ < πn

θ < πn

Combine the intervals

θ > πn - \frac{π}{2} and θ < πn

Merge Overlapping Intervals

- \frac{π}{2} + πn < θ < πn

Combine the intervals

False for all θ \in R and - \frac{π}{2} + πn < θ < πn

Merge Overlapping Intervals

False for all θ \in R

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cos(θ)=45\land 0<θ<90,sin(θ)

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

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arcsec(-sqrt(2))0<= x<= 2pi

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