What is the general solution for 4cot(θ)sin(θ)=2 ?

The general solution for 4cot(θ)sin(θ)=2 is θ= pi/3+2pin,θ=(5pi)/3+2pin

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4cot(θ)sin(θ)=2

4 cot (θ) sin (θ) = 2

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

4 cot (θ) sin (θ) = 2

Subtract

2

from both sides

4 cot (θ) sin (θ) - 2 = 0

Rewrite using trig identities

- 2 + 4 cot (θ) sin (θ)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - 2 + 4 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ)

4 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ) = 4 cos (θ)

4 \cdot \frac{cos ( θ )}{sin ( θ )} sin (θ)

Multiply fractions:

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

= \frac{cos ( θ ) \cdot 4 sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= cos (θ) \cdot 4

= - 2 + 4 cos (θ)

- 2 + 4 cos (θ) = 0

Move

2

to the right side

- 2 + 4 cos (θ) = 0

Add

2

to both sides

- 2 + 4 cos (θ) + 2 = 0 + 2

Simplify

4 cos (θ) = 2

4 cos (θ) = 2

Divide both sides by

4

4 cos (θ) = 2

Divide both sides by

4

\frac{4 cos ( θ )}{4} = \frac{2}{4}

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

2 sin (2 x) - 5 tan (2 x) = 0

75 = \frac{120}{4} \cdot cos (\frac{2 \cdot π \cdot x}{360}) + \frac{120}{2}

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

3 sin (x) + cos (x) = 2

15 cos^{2} (x) - 4 cos (x) - 3 = 0

What is the general solution for 4cot(θ)sin(θ)=2 ?
The general solution for 4cot(θ)sin(θ)=2 is θ= pi/3+2pin,θ=(5pi)/3+2pin