What is the general solution for (csc(θ))/(cot(θ))=-2tan(θ) ?

The general solution for (csc(θ))/(cot(θ))=-2tan(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin

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(csc(θ))/(cot(θ))=-2tan(θ)

\frac{csc ( θ )}{cot ( θ )} = - 2 tan (θ)

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Degrees

θ = 21 0^{\circ} + 36 0^{\circ} n, θ = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{csc ( θ )}{cot ( θ )} = - 2 tan (θ)

Subtract

- 2 tan (θ)

from both sides

\frac{csc ( θ )}{cot ( θ )} + 2 tan (θ) = 0

Simplify

\frac{csc ( θ )}{cot ( θ )} + 2 tan (θ) : \frac{csc ( θ ) + 2 tan ( θ ) cot ( θ )}{cot ( θ )}

\frac{csc ( θ )}{cot ( θ )} + 2 tan (θ)

Convert element to fraction:

2 tan (θ) = \frac{2 tan ( θ ) cot ( θ )}{cot ( θ )}

= \frac{csc ( θ )}{cot ( θ )} + \frac{2 tan ( θ ) cot ( θ )}{cot ( θ )}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{csc ( θ ) + 2 tan ( θ ) cot ( θ )}{cot ( θ )}

\frac{csc ( θ ) + 2 tan ( θ ) cot ( θ )}{cot ( θ )} = 0

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

csc (θ) + 2 tan (θ) cot (θ) = 0

Rewrite using trig identities

csc (θ) + 2 cot (θ) tan (θ)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

= csc (θ) + 2 cot (θ) \frac{1}{cot ( θ )}

2 cot (θ) \frac{1}{cot ( θ )} = 2

2 cot (θ) \frac{1}{cot ( θ )}

Multiply fractions:

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

= \frac{1 \cdot 2 cot ( θ )}{cot ( θ )}

Cancel the common factor:

cot (θ)

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= csc (θ) + 2

2 + csc (θ) = 0

Move

2

to the right side

2 + csc (θ) = 0

Subtract

2

from both sides

2 + csc (θ) - 2 = 0 - 2

Simplify

csc (θ) = - 2

csc (θ) = - 2

General solutions for

csc (θ) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

2 cos^{2} (θ) + 3 sin (θ) = 0

sin (2 θ) + 3 cos (θ) = 0

2 cot (x) + csc^{2} (x) = 0

2 cos^{2} (θ) + 3 cos (θ) + 4 = 3

2 sin (3 θ) cos (2 θ) - cos (2 θ) = 0

What is the general solution for (csc(θ))/(cot(θ))=-2tan(θ) ?
The general solution for (csc(θ))/(cot(θ))=-2tan(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin