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Popular Trigonometry >

csc(θ)((sqrt(26))/5)tan(θ)<0

Solution

csc (θ) (\frac{26}{5}) tan (θ) < 0

Solution

\frac{π}{2} + 2 πn < θ < \frac{3 π}{2} + 2 πn

Interval Notation

(\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn)

Decimal

1.57079 \dots + 2 πn < θ < 4.71238 \dots + 2 πn

Solution steps

csc (θ) \frac{26}{5} tan (θ) < 0

Periodicity of

csc (θ) \frac{26}{5} tan (θ) : 2 π

csc (θ) \frac{26}{5} tan (θ)

is composed of the following functions and periods:

csc (θ)

with periodicity of

2 π

The compound periodicity is:

= 2 π

Express with sin, cos

csc (θ) \frac{26}{5} tan (θ) < 0

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

\frac{1}{sin ( θ )} \cdot \frac{26}{5} tan (θ) < 0

Use the basic trigonometric identity:

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

\frac{1}{sin ( θ )} \cdot \frac{26}{5} \cdot \frac{sin ( θ )}{cos ( θ )} < 0

\frac{1}{sin ( θ )} \cdot \frac{26}{5} \cdot \frac{sin ( θ )}{cos ( θ )} < 0

Simplify

\frac{1}{sin ( θ )} \cdot \frac{26}{5} \cdot \frac{sin ( θ )}{cos ( θ )} : \frac{26}{5 cos ( θ )}

\frac{1}{sin ( θ )} \cdot \frac{26}{5} \cdot \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{1 \cdot 26 sin ( θ )}{sin ( θ ) \cdot 5 cos ( θ )}

Cancel the common factor:

sin (θ)

= \frac{1 \cdot 26}{5 cos ( θ )}

Multiply:

1 \cdot 26 = 26

= \frac{26}{5 cos ( θ )}

\frac{26}{5 cos ( θ )} < 0

Find the zeroes and undifined points of

\frac{26}{5 cos ( θ )}

for

0 \leq θ < 2 π

To find the zeroes, set the inequality to zero

\frac{26}{5 cos ( θ )} = 0

\frac{26}{5 cos ( θ )} = 0 :

No Solution for

θ \in R

\frac{26}{5 cos ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

26 = 0

The sides are not equal

No Solution for θ \in R

Find the undefined points:

θ = \frac{π}{2}, θ = \frac{3 π}{2}

Find the zeros of the denominator

5 cos (θ) = 0

Divide both sides by

5

5 cos (θ) = 0

Divide both sides by

5

\frac{5 cos ( θ )}{5} = \frac{0}{5}

Simplify

cos (θ) = 0

cos (θ) = 0

General solutions for

cos (θ) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

Solutions for the range

0 \leq θ < 2 π

θ = \frac{π}{2}, θ = \frac{3 π}{2}

\frac{π}{2}, \frac{3 π}{2}

Identify the intervals

0 < θ < \frac{π}{2}, \frac{π}{2} < θ < \frac{3 π}{2}, \frac{3 π}{2} < θ < 2 π

Summarize in a table:

26 cos (θ) \frac{26}{5 c o s ( θ )} θ = 0 + + + 0 < θ < \frac{π}{2} + + + θ = \frac{π}{2} + 0 Undefined \frac{π}{2} < θ < \frac{3 π}{2} + - - θ = \frac{3 π}{2} + 0 Undefined \frac{3 π}{2} < θ < 2 π + + + θ = 2 π + + +

Identify the intervals that satisfy the required condition:

< 0

\frac{π}{2} < θ < \frac{3 π}{2}

Apply the periodicity of

csc (θ) \frac{26}{5} tan (θ)

\frac{π}{2} + 2 πn < θ < \frac{3 π}{2} + 2 πn

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