What is the general solution for 3-3sin(t)=3sqrt(3)cos(t) ?

The general solution for 3-3sin(t)=3sqrt(3)cos(t) is t= pi/2+2pin,t=(11pi)/6+2pin

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3-3sin(t)=3sqrt(3)cos(t)

3 - 3 sin (t) = 33 cos (t)

t = \frac{π}{2} + 2 πn, t = \frac{11 π}{6} + 2 πn

Solution steps

3 - 3 sin (t) = 33 cos (t)

Square both sides

(3 - 3 sin (t))^{2} = (33 cos (t))^{2}

Subtract

(33 cos (t))^{2}

from both sides

(3 - 3 sin (t))^{2} - 27 cos^{2} (t) = 0

Rewrite using trig identities

- 18 - 18 sin (t) + 36 sin^{2} (t) = 0

Solve by substitution

sin (t) = 1, sin (t) = - \frac{1}{2}

sin (t) = 1 : t = \frac{π}{2} + 2 πn

sin (t) = - \frac{1}{2} : t = \frac{7 π}{6} + 2 πn, t = \frac{11 π}{6} + 2 πn

Combine all the solutions

t = \frac{π}{2} + 2 πn, t = \frac{7 π}{6} + 2 πn, t = \frac{11 π}{6} + 2 πn

Verify solutions by plugging them into the original equation

t = \frac{π}{2} + 2 πn, t = \frac{11 π}{6} + 2 πn

sin (θ) = \frac{3}{10}

30 arctan (x) = 5 π

cosh (2 x) - 10 cosh (x) + 13 = 0

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

7 cos^{2} (x) + 14 cos (x) + 7 = 0

What is the general solution for 3-3sin(t)=3sqrt(3)cos(t) ?
The general solution for 3-3sin(t)=3sqrt(3)cos(t) is t= pi/2+2pin,t=(11pi)/6+2pin