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

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

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

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

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

Degrees

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

Solution steps

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

Rewrite using trig identities

sin (2 t) + 2 cos (t)

Use the Double Angle identity:

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

= 2 sin (t) cos (t) + 2 cos (t)

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

Factor

2 cos (t) + 2 cos (t) sin (t) : 2 cos (t) (sin (t) + 1)

2 cos (t) + 2 cos (t) sin (t)

Factor out common term

2 cos (t)

= 2 cos (t) (1 + sin (t))

2 cos (t) (sin (t) + 1) = 0

Solving each part separately

cos (t) = 0 or sin (t) + 1 = 0

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

cos (t) = 0

General solutions for

cos (t) = 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}

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

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

sin (t) + 1 = 0 : t = \frac{3 π}{2} + 2 πn

sin (t) + 1 = 0

Move

1

to the right side

sin (t) + 1 = 0

Subtract

1

from both sides

sin (t) + 1 - 1 = 0 - 1

Simplify

sin (t) = - 1

sin (t) = - 1

General solutions for

sin (t) = - 1

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

t = \frac{3 π}{2} + 2 πn

t = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

tan (θ) = cos (θ)

cos (x) = 0.766

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

sin (θ) = - sin (θ) cos (θ)

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

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