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

The general solution for (cot(θ))/(csc(θ))=sin(θ) is θ= pi/4+pin

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(cot(θ))/(csc(θ))=sin(θ)

\frac{cot ( θ )}{csc ( θ )} = sin (θ)

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

\frac{cot ( θ )}{csc ( θ )} = sin (θ)

Subtract

sin (θ)

from both sides

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

Simplify

\frac{cot ( θ )}{csc ( θ )} - sin (θ) : \frac{cot ( θ ) - sin ( θ ) csc ( θ )}{csc ( θ )}

\frac{cot ( θ )}{csc ( θ )} - sin (θ)

Convert element to fraction:

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

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

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

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

cot (θ) - sin (θ) csc (θ) = 0

Rewrite using trig identities

cot (θ) - sin (θ) csc (θ)

csc (θ) sin (θ) = 1

csc (θ) sin (θ)

Express with sin, cos

csc (θ) sin (θ)

Use the basic trigonometric identity:

csc (θ) = \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )} sin (θ)

\frac{1}{sin ( θ )} sin (θ) = 1

\frac{1}{sin ( θ )} sin (θ)

Multiply fractions:

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

= \frac{1 sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= 1

= 1

= cot (θ) - 1

cot (θ) - 1 = 0

Move

1

to the right side

cot (θ) - 1 = 0

Add

1

to both sides

cot (θ) - 1 + 1 = 0 + 1

Simplify

cot (θ) = 1

cot (θ) = 1

General solutions for

cot (θ) = 1

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

cos (t) = - \frac{2}{3}

sin (θ) = 1 + (1 - (cos (θ) - 1)^{2})

0 = 4 cos (4 x)

tan (θ) = \frac{8}{7}, π < θ < \frac{3 π}{2}

csc (θ) sin (θ) = cot (θ)

What is the general solution for (cot(θ))/(csc(θ))=sin(θ) ?
The general solution for (cot(θ))/(csc(θ))=sin(θ) is θ= pi/4+pin