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

The general solution for cos(x)sin(x)+2cos^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=-1.10714…+pin

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

cos (x) sin (x) + 2 cos^{2} (x) = 0

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 1.10714 \dots + πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

cos (x) sin (x) + 2 cos^{2} (x) = 0

Factor

cos (x) sin (x) + 2 cos^{2} (x) : cos (x) (sin (x) + 2 cos (x))

cos (x) sin (x) + 2 cos^{2} (x)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

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

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

Factor out common term

cos (x)

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

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

Solving each part separately

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

sin (x) + 2 cos (x) = 0 : x = arctan (- 2) + πn

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

Rewrite using trig identities

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

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + 2 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

\frac{sin ( x )}{cos ( x )} + 2 = 0

Use the basic trigonometric identity:

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

tan (x) + 2 = 0

tan (x) + 2 = 0

Move

2

to the right side

tan (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 1.10714 \dots + πn

tan (x) = \frac{130}{45}

csc (\frac{π}{42} x) = 1

arctan (4 - 2 x) = arctan (2 x)

tan (x) = tan (5 0^{\circ})

tan (θ) = 3, 0 < x < 2 π

What is the general solution for cos(x)sin(x)+2cos^2(x)=0 ?
The general solution for cos(x)sin(x)+2cos^2(x)=0 is x= pi/2+2pin,x=(3pi)/2+2pin,x=-1.10714…+pin