What is the general solution for sin(2x)+6cos(x)=0 ?

The general solution for sin(2x)+6cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin

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sin(2x)+6cos(x)=0

sin (2 x) + 6 cos (x) = 0

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

Solution steps

sin (2 x) + 6 cos (x) = 0

Rewrite using trig identities

6 cos (x) + 2 cos (x) sin (x) = 0

Factor

6 cos (x) + 2 cos (x) sin (x) : 2 cos (x) (sin (x) + 3)

2 cos (x) (sin (x) + 3) = 0

Solving each part separately

cos (x) = 0 or sin (x) + 3 = 0

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

sin (x) + 3 = 0 :

No Solution

Combine all the solutions

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

cos (3 x) = \frac{1}{2}, 0 \leq x \leq 2 π

2 cos^{2} (x) - cos (x) = 0, 0 \leq x < 2 π

- 2 cos^{2} (x) = cos (x)

14 sin^{2} (x) = 7

csc (x) = sin (x)

What is the general solution for sin(2x)+6cos(x)=0 ?
The general solution for sin(2x)+6cos(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin