What is the general solution for cos(θ)sin(θ)-4cos(θ)=0 ?

The general solution for cos(θ)sin(θ)-4cos(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin

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cos(θ)sin(θ)-4cos(θ)=0

cos (θ) sin (θ) - 4 cos (θ) = 0

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

Degrees

θ = 9 0^{\circ} + 36 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

cos (θ) sin (θ) - 4 cos (θ) = 0

Factor

cos (θ) sin (θ) - 4 cos (θ) : cos (θ) (sin (θ) - 4)

cos (θ) sin (θ) - 4 cos (θ)

Factor out common term

cos (θ)

= cos (θ) (sin (θ) - 4)

cos (θ) (sin (θ) - 4) = 0

Solving each part separately

cos (θ) = 0 or sin (θ) - 4 = 0

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

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

sin (θ) - 4 = 0 :

No Solution

sin (θ) - 4 = 0

Move

4

to the right side

sin (θ) - 4 = 0

Add

4

to both sides

sin (θ) - 4 + 4 = 0 + 4

Simplify

sin (θ) = 4

sin (θ) = 4

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

sec (x) sin (x) = 2 sin (x)

cot (A) - 1 = 0

tan^{2} (x) = - 3

cot^{2} (x) = 3 cot (x)

tan (θ) = \frac{4}{3}, π < θ < \frac{3 π}{2}

What is the general solution for cos(θ)sin(θ)-4cos(θ)=0 ?
The general solution for cos(θ)sin(θ)-4cos(θ)=0 is θ= pi/2+2pin,θ=(3pi)/2+2pin