What is the general solution for (tan(θ)cos(θ))/(csc(θ))=sin(θ) ?

The general solution for (tan(θ)cos(θ))/(csc(θ))=sin(θ) is No Solution for θ\in\mathbb{R}

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(tan(θ)cos(θ))/(csc(θ))=sin(θ)

\frac{tan ( θ ) cos ( θ )}{csc ( θ )} = sin (θ)

No Solution for θ \in R

Solution steps

\frac{tan ( θ ) cos ( θ )}{csc ( θ )} = sin (θ)

Subtract

sin (θ)

from both sides

\frac{tan ( θ ) cos ( θ )}{csc ( θ )} - sin (θ) = 0

Simplify

\frac{tan ( θ ) cos ( θ )}{csc ( θ )} - sin (θ) : \frac{tan ( θ ) cos ( θ ) - sin ( θ ) csc ( θ )}{csc ( θ )}

\frac{tan ( θ ) cos ( θ )}{csc ( θ )} - sin (θ)

Convert element to fraction:

sin (θ) = \frac{sin ( θ ) csc ( θ )}{csc ( θ )}

= \frac{tan ( θ ) cos ( θ )}{csc ( θ )} - \frac{sin ( θ ) csc ( θ )}{csc ( θ )}

Since the denominators are equal, combine the fractions:

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

= \frac{tan ( θ ) cos ( θ ) - sin ( θ ) csc ( θ )}{csc ( θ )}

\frac{tan ( θ ) cos ( θ ) - sin ( θ ) csc ( θ )}{csc ( θ )} = 0

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

tan (θ) cos (θ) - sin (θ) csc (θ) = 0

Rewrite using trig identities

cos (θ) tan (θ) - 1

cos (θ) tan (θ) = sin (θ)

cos (θ) tan (θ)

Use the basic trigonometric identity:

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

= cos (θ) \frac{sin ( θ )}{cos ( θ )}

Simplify

cos (θ) \frac{sin ( θ )}{cos ( θ )} : sin (θ)

cos (θ) \frac{sin ( θ )}{cos ( θ )}

Multiply fractions:

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

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

Cancel the common factor:

cos (θ)

= sin (θ)

= sin (θ)

= - 1 + sin (θ)

- 1 + sin (θ) = 0

Move

1

to the right side

- 1 + sin (θ) = 0

Add

1

to both sides

- 1 + sin (θ) + 1 = 0 + 1

Simplify

sin (θ) = 1

sin (θ) = 1

General solutions for

sin (θ) = 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

θ = \frac{π}{2} + 2 πn

θ = \frac{π}{2} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn

No Solution for θ \in R

sin (θ) = \frac{44}{12}

cos (A) = \frac{2 ^{2} + 2.2 4 ^{2} - 2.2 4 ^{2}}{2 ( 2 ) ( 2.24 )}

cos (θ) = - \frac{3}{5}, θ, 3

sin (3 x) = 0.5

cos^{2} (x) - 5 sin (x) + 5 = 0

What is the general solution for (tan(θ)cos(θ))/(csc(θ))=sin(θ) ?
The general solution for (tan(θ)cos(θ))/(csc(θ))=sin(θ) is No Solution for θ\in\mathbb{R}