Determine the general Solution of the forlowing 1. \( \cos (2 x+30)=-\sin \left(x-20^{\circ}\right) \) 2. \( \cos \left(\theta-45^{\circ}\right)=\sin (30-\theta) \) 3. \( \sin (2 x+45)=\cos 30^{\circ} \) 4. \( \sin 2 x=-\cos 30^{\circ} \) 5. \( \cos \left(3 x-30^{\circ}\right)=\sin \left(x+60^{\circ}\right) \)

To solve these trigonometric equations, let’s have some fun with each one! 1. **Get ready for some angle adjustments in the cosine world!** First, we convert \(-\sin\) to \(\cos\) using the identity \(-\sin(θ) = \cos(θ + 90^\circ)\), leading to the equation \(\cos(2x + 30^\circ) = \cos(x - 20^\circ + 90^\circ)\). This gives us \(2x + 30^\circ = 360k \pm (x + 70^\circ)\), leading to \(x = 360k + 40^\circ\) or \(x = 360k + 100^\circ\) for all integers \(k\). 2. **Let's work with some angles!** The symmetry of sine and cosine helps us here too! Convert the equation using \(\cos(θ) = \sin(90^\circ - θ)\). This gives us \(\sin(30 - θ) = \sin(90^\circ - θ + 45^\circ)\). Solving the resulting equations, we find the general solutions are of the form \(\theta = 30^\circ + 180k\) or \(\theta = 75^\circ + 180k\) for integers \(k\). 3. **Double the x, double the fun!** For the third equation, we rewrite \(\cos 30^\circ\) as \(\frac{\sqrt{3}}{2}\). This turns into \(\sin(2x + 45^\circ) = \frac{\sqrt{3}}{2}\). The solutions are found from the unit circle, leading to \(2x + 45^\circ = 60^\circ + 360k\) or \(2x + 45^\circ = 120^\circ + 360k\), which simplifies to \(x = \frac{15^\circ + 180k}{2}\) or \(x = \frac{75^\circ + 180k}{2}\). 4. **Keep the angles coming!** Set \(\sin 2x = -\frac{\sqrt{3}}{2}\) since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). The solutions are then \(2x = 240^\circ + 360k\) or \(2x = 300^\circ + 360k\), leading to \(x = 120^\circ + 180k\) or \(x = 150^\circ + 180k\). 5. **Let’s wrap up with some counting!** Interchange \(\sin\) and \(\cos\) using angle co-function identities, giving us \(\cos(3x - 30^\circ) = \sin(90^\circ - (x + 60^\circ))\). Working through these identities ends up giving solutions of the form \(3x - 30^\circ = 30^\circ + 360k\) or \(3x - 30^\circ = 150^\circ + 360k\), yielding \(x = 20^\circ + 120k\) or \(x = 60^\circ + 120k\). If you keep a sharp eye on those angles, solving for x becomes a walk in the park! Happy calculating!