What is the general solution for 4sin(x)cos(x)+5cos(x)=0 ?

The general solution for 4sin(x)cos(x)+5cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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4sin(x)cos(x)+5cos(x)=0

4 sin (x) cos (x) + 5 cos (x) = 0

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

Degrees

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

Solution steps

4 sin (x) cos (x) + 5 cos (x) = 0

Factor

4 sin (x) cos (x) + 5 cos (x) : cos (x) (4 sin (x) + 5)

4 sin (x) cos (x) + 5 cos (x)

Factor out common term

cos (x)

= cos (x) (4 sin (x) + 5)

cos (x) (4 sin (x) + 5) = 0

Solving each part separately

cos (x) = 0 or 4 sin (x) + 5 = 0

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

cos (x) = 0

General solutions for

cos (x) = 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}

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

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

4 sin (x) + 5 = 0 :

No Solution

4 sin (x) + 5 = 0

Move

5

to the right side

4 sin (x) + 5 = 0

Subtract

5

from both sides

4 sin (x) + 5 - 5 = 0 - 5

Simplify

4 sin (x) = - 5

4 sin (x) = - 5

Divide both sides by

4

4 sin (x) = - 5

Divide both sides by

4

\frac{4 sin ( x )}{4} = \frac{- 5}{4}

Simplify

sin (x) = - \frac{5}{4}

sin (x) = - \frac{5}{4}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

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

2 sin (4 x) = 1

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

(cos (x) + 1) (cos (x) - 4) = 0

cos (10 x) = 0

7 - 5 cos (x) = 10

What is the general solution for 4sin(x)cos(x)+5cos(x)=0 ?
The general solution for 4sin(x)cos(x)+5cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin