What is the general solution for (e^{-ln(-(sin(θ))/(cos(θ)))})/2*sin(θ)=0 ?

The general solution for (e^{-ln(-(sin(θ))/(cos(θ)))})/2*sin(θ)=0 is No Solution for θ\in\mathbb{R}

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(e^{-ln(-(sin(θ))/(cos(θ)))})/2*sin(θ)=0

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2} \cdot sin (θ) = 0

No Solution for θ \in R

Solution steps

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2} sin (θ) = 0

Simplify

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2} sin (θ) : - \frac{cos ( θ )}{2}

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2} sin (θ)

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2} = - \frac{cos ( θ )}{2 sin ( θ )}

\frac{e ^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}}{2}

e^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})} = - \frac{cos ( θ )}{sin ( θ )}

e^{- l n (- \frac{s i n ( θ )}{c o s ( θ )})}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (e^{l n (- \frac{s i n ( θ )}{c o s ( θ )})})^{- 1}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (- \frac{s i n ( θ )}{c o s ( θ )})} = - \frac{sin ( θ )}{cos ( θ )}

= (- \frac{sin ( θ )}{cos ( θ )})^{- 1}

Apply exponent rule:

a^{- 1} = \frac{1}{a}

= \frac{1}{- \frac{s i n ( θ )}{c o s ( θ )}}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{1}{\frac{s i n ( θ )}{c o s ( θ )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

\frac{1}{\frac{s i n ( θ )}{c o s ( θ )}} = \frac{cos ( θ )}{sin ( θ )}

= - \frac{cos ( θ )}{sin ( θ )}

= \frac{- \frac{c o s ( θ )}{s i n ( θ )}}{2}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{c o s ( θ )}{s i n ( θ )}}{2}

Apply the fraction rule:

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

\frac{\frac{c o s ( θ )}{s i n ( θ )}}{2} = \frac{cos ( θ )}{sin ( θ ) \cdot 2}

= - \frac{cos ( θ )}{sin ( θ ) \cdot 2}

= - \frac{cos ( θ )}{2 sin ( θ )} sin (θ)

Multiply fractions:

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

= - \frac{cos ( θ ) sin ( θ )}{sin ( θ ) \cdot 2}

Cancel the common factor:

sin (θ)

= - \frac{cos ( θ )}{2}

- \frac{cos ( θ )}{2} = 0

Multiply both sides by

2

- \frac{cos ( θ )}{2} = 0

Multiply both sides by

2

2 (- \frac{cos ( θ )}{2}) = 0 \cdot 2

Simplify

- cos (θ) = 0

- cos (θ) = 0

Divide both sides by

- 1

- cos (θ) = 0

Divide both sides by

- 1

\frac{- cos ( θ )}{- 1} = \frac{0}{- 1}

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

Since the equation is undefined for:

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

No Solution for θ \in R

7 sin (2 x) + 12 sin (x) = 0

x, cos (x) = cos (2 x)

cos (x) = 0.50074242

2 sin (4 x) - 1 = 0, - 2 π \leq x \leq 3 π

tan (θ) = \frac{10}{10}

What is the general solution for (e^{-ln(-(sin(θ))/(cos(θ)))})/2*sin(θ)=0 ?
The general solution for (e^{-ln(-(sin(θ))/(cos(θ)))})/2*sin(θ)=0 is No Solution for θ\in\mathbb{R}