Determino the genoral Solution of the following. 1. $$ \sin 2 heta=\sin ( heta-30) $$ 2. $$ \cos (x-10)=\cos (2 x+15) $$ 3. $$ an 3 heta= an (2 heta+45) $$ 4. $$ \sin \left(2 heta+60^{\circ} ight)=-\sin heta $$ 5. $$ \cos 4 x=\cos \left(2 x-30^{\circ} ight) $$

Question

Determino the genoral Solution of the following. 1. $$ \sin 2 	heta=\sin (	heta-30) $$ 2. $$ \cos (x-10)=\cos (2 x+15) $$ 3. $$ 	an 3 	heta=	an (2 	heta+45) $$ 4. $$ \sin \left(2 	heta+60^{\circ}ight)=-\sin 	heta $$ 5. $$ \cos 4 x=\cos \left(2 x-30^{\circ}ight) $$

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Solve the equation by following steps:
- step0: Solve for $$\theta$$:
  $$\theta -30-2\theta =k\times \frac{\pi }{2}$$
- step1: Subtract the terms:
  $$-\theta -30=\frac{\pi }{2}k$$
- step2: Move the constant to the right side:
  $$-\theta =\frac{\pi }{2}k+30$$
- step3: Change the signs:
  $$\theta =-\frac{\pi }{2}k-30$$
Solve the equation $$ 4x - (2x - 30) = k2\pi $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$4x-\left(2x-30\right)=k\times 2\pi $$
- step1: Subtract the terms:
  $$2x+30=2\pi k$$
- step2: Move the constant to the right side:
  $$2x=2\pi k-30$$
- step3: Rewrite the expression:
  $$2x=2k\pi -30$$
- step4: Divide both sides:
  $$\frac{2x}{2}=\frac{2k\pi -30}{2}$$
- step5: Divide the numbers:
  $$x=k\pi -15$$
Solve the equation $$ (x-10) - (2x+15) = k2\pi $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\left(x-10\right)-\left(2x+15\right)=k\times 2\pi $$
- step1: Simplify:
  $$-x-25=2\pi k$$
- step2: Move the constant to the right side:
  $$-x=2\pi k+25$$
- step3: Change the signs:
  $$x=-2\pi k-25$$
- step4: Simplify:
  $$x=-2k\pi -25$$
Solve the equation $$ 3\theta - (2\theta + 45) = k\pi $$.
Solve the equation by following steps:
- step0: Solve for $$\theta$$:
  $$3\theta -\left(2\theta +45\right)=k\pi $$
- step1: Subtract the terms:
  $$\theta -45=\pi k$$
- step2: Move the constant to the right side:
  $$\theta =\pi k+45$$
- step3: Simplify:
  $$\theta =k\pi +45$$
Solve the equation $$ \frac{1}{2} + 60 + \theta = k\pi $$.
Solve the equation by following steps:
- step0: Solve for $$\theta$$:
  $$\frac{1}{2}+60+\theta =k\pi $$
- step1: Add the numbers:
  $$\frac{121}{2}+\theta =\pi k$$
- step2: Move the constant to the right side:
  $$\theta =\pi k-\frac{121}{2}$$
- step3: Rewrite the expression:
  $$\theta =k\pi -\frac{121}{2}$$
Here are the general solutions for each of the equations provided:

1. For the equation $$ \sin 2 \theta = \sin (\theta - 30) $$:
   $$
   \theta = -\frac{\pi}{2}k - 30
   $$
   where $$ k $$ is any integer.

2. For the equation $$ \cos (x - 10) = \cos (2x + 15) $$:
   $$
   x = -2k\pi - 25
   $$
   where $$ k $$ is any integer.

3. For the equation $$ \tan 3\theta = \tan (2\theta + 45) $$:
   $$
   \theta = k\pi + 45
   $$
   where $$ k $$ is any integer.

4. For the equation $$ \sin(2\theta + 60^{\circ}) = -\sin \theta $$:
   $$
   \theta = k\pi - \frac{121}{2}
   $$
   where $$ k $$ is any integer.

5. For the equation $$ \cos 4x = \cos(2x - 30^{\circ}) $$:
   $$
   x = k\pi - 15
   $$
   where $$ k $$ is any integer.

These solutions represent the general forms for each trigonometric equation.
Here are the general solutions for each equation: 1. $$ \theta = -\frac{\pi}{2}k - 30 $$ 2. $$ x = -2k\pi - 25 $$ 3. $$ \theta = k\pi + 45 $$ 4. $$ \theta = k\pi - \frac{121}{2} $$ 5. $$ x = k\pi - 15 $$ where $$ k $$ is any integer.

Determino the genoral Solution of the following. 1. \( \sin 2 \theta=\sin (\theta-30) \) 2. \( \cos (x-10)=\cos (2 x+15) \) 3. \( \tan 3 \theta=\tan (2 \theta+45) \) 4. \( \sin \left(2 \theta+60^{\circ}\right)=-\sin \theta \) 5. \( \cos 4 x=\cos \left(2 x-30^{\circ}\right) \)

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