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

The general solution for sin(2t)=cos(t) is t= pi/2+2pin,t=(3pi)/2+2pin,t= pi/6+2pin,t=(5pi)/6+2pin

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sin(2t)=cos(t)

sin (2 t) = cos (t)

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

Solution steps

sin (2 t) = cos (t)

Subtract

cos (t)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

cos (t) = 0 or 2 sin (t) - 1 = 0

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

2 sin (t) - 1 = 0 : t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

Combine all the solutions

t = \frac{π}{2} + 2 πn, t = \frac{3 π}{2} + 2 πn, t = \frac{π}{6} + 2 πn, t = \frac{5 π}{6} + 2 πn

cos (2 x) + 1 = cos (x)

4 cos (x) + 1 = 0

2 cot^{2} (x) + csc^{2} (x) - 2 = 0

cos^{2} (x) = \frac{1}{2} sin (2 x)

3 cot (θ) + 5 = 0

What is the general solution for sin(2t)=cos(t) ?
The general solution for sin(2t)=cos(t) is t= pi/2+2pin,t=(3pi)/2+2pin,t= pi/6+2pin,t=(5pi)/6+2pin