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

The general solution for (2cos(θ))/(tan(θ))=cot(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin

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(2cos(θ))/(tan(θ))=cot(θ)

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

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

cot (θ)

from both sides

\frac{2 cos ( θ )}{tan ( θ )} - cot (θ) = 0

Simplify

\frac{2 cos ( θ )}{tan ( θ )} - cot (θ) : \frac{2 cos ( θ ) - cot ( θ ) tan ( θ )}{tan ( θ )}

\frac{2 cos ( θ )}{tan ( θ )} - cot (θ)

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

\frac{2 cos ( θ ) - cot ( θ ) tan ( θ )}{tan ( θ )} = 0

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

2 cos (θ) - cot (θ) tan (θ) = 0

Rewrite using trig identities

2 cos (θ) - cot (θ) tan (θ)

Use the basic trigonometric identity:

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

= 2 cos (θ) - cot (θ) \frac{1}{cot ( θ )}

cot (θ) \frac{1}{cot ( θ )} = 1

cot (θ) \frac{1}{cot ( θ )}

Multiply fractions:

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

= \frac{1 \cdot cot ( θ )}{cot ( θ )}

Cancel the common factor:

cot (θ)

= 1

= 2 cos (θ) - 1

- 1 + 2 cos (θ) = 0

Move

1

to the right side

- 1 + 2 cos (θ) = 0

Add

1

to both sides

- 1 + 2 cos (θ) + 1 = 0 + 1

Simplify

2 cos (θ) = 1

2 cos (θ) = 1

Divide both sides by

2

2 cos (θ) = 1

Divide both sides by

2

\frac{2 cos ( θ )}{2} = \frac{1}{2}

Simplify

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \frac{1}{2}

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}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

sin^{4} (x) = 0

tan (θ) = \frac{7}{24}

- 22 = - 2 sec (x + 15 0^{\circ})

9 sin (θ) + 11 = 0

(4 sin (2 x) + 4 cos (2 x))^{2} = 16

What is the general solution for (2cos(θ))/(tan(θ))=cot(θ) ?
The general solution for (2cos(θ))/(tan(θ))=cot(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin