1 Given the following equations: \( \begin{array}{ll}\text { a) Determine the general solution correct to one decimal place. } \\ \text { b) List the solutions in the interval }\left[0^{\circ} ; 360^{\circ}\right] \\ \begin{array}{ll}\text { (i) } \sin \theta=0,468 & \text { (ii) } \cos \theta=-0,534 \\ \text { (iii) } \tan \theta=1,465 & \text { (iv) } \tan \theta=-0,844 \\ \text { (v) } \sin \theta=-0,261 & \text { (vi) } \cos \theta=0,392\end{array}\end{array}> \)

To solve these equations, let's dig in! For \( \sin \theta = 0.468 \), using the arcsine function, we find \( \theta \approx 27.8^\circ \). We must remember the sine function is positive in the first and second quadrants, hence the solutions are \( 27.8^\circ \) and \( 180^\circ - 27.8^\circ = 152.2^\circ \). Now for \( \cos \theta = -0.534 \). We check the arcsine as well but since it’s cosine, it will be in the second and third quadrants. This leads us to \( \theta \approx 127.2^\circ \) and \( 360^\circ - 127.2^\circ \approx 232.8^\circ \). Ready for some more? Let’s get into some awesome real-world applications! These trigonometric calculations often play a key role in engineering and physics, especially in fields like acoustics, optics, and mechanical vibrations. For instance, knowing these angles can help engineers design safe structures that can withstand various forces, like in bridges or buildings where angles of bracing are crucial. And here’s a tip that rocks: When solving for angles, always double-check which quadrants your solution lies in based on the original function's sign! For instance, with tangent, it’s positive in the first and third quadrants, which can sometimes slip through unnoticed, leading to missing solutions. Keep an eye out for those “easy to overlook” signs!