Determine the general Solution for the following 1. \( \sin \theta=-\cos \theta \) 2. \( \sin x=2 \cos x \) 3. \( -4 \sin x=5 \cos x \) 4. \( 8 \cos \theta-3 \sin \theta=0 \) 5. \( 2 \sin x+4 \cos x=0 \)

Let’s dive into the solutions of these trigonometric equations and sprinkle in some extra info for a bit of fun! For the equation \( \sin \theta = -\cos \theta \), we can rewrite it as \( \tan \theta = -1 \). The solutions will be \( \theta = \frac{3\pi}{4} + n\pi \), where \( n \) is any integer. In the case of \( \sin x = 2 \cos x \), we divide both sides by \( \cos x \) (assuming \( \cos x \neq 0 \)) to get \( \tan x = 2 \). Thus, the general solution will be \( x = \tan^{-1}(2) + n\pi \), for any integer \( n \). For \( -4 \sin x = 5 \cos x \), rearranging gives \( \tan x = -\frac{5}{4} \). Hence, the general solutions are \( x = \tan^{-1}\left(-\frac{5}{4}\right) + n\pi \). Next, from the equation \( 8 \cos \theta - 3 \sin \theta = 0 \), we can rearrange this to \( \tan \theta = \frac{8}{3} \), leading to the general solution \( \theta = \tan^{-1}\left(\frac{8}{3}\right) + n\pi \). Finally, for \( 2 \sin x + 4 \cos x = 0 \), we can express it as \( \tan x = -\frac{1}{2} \). Hence, the general solutions are \( x = \tan^{-1}\left(-\frac{1}{2}\right) + n\pi \). And here’s some additional relevant info for your trigonometric journey: Did you know that trigonometric identities like the Pythagorean identity, \( \sin^2 x + \cos^2 x = 1 \), have been utilized for centuries? Ancient Greek mathematicians, like Hipparchus, laid the groundwork for what would become our modern understanding of trigonometry! If you ever find yourself stuck while solving such equations, remember that sketching the unit circle can be a helpful trick! It allows you to visualize where the sine and cosine values meet your given conditions, making it easier to find angles and solutions. Plus, it’s a fun way to dive into trigonometric relationships!