Trigonometry TRIGONOMETRIC IDENTITIES The tangent function, \( f(\alpha)=\tan \alpha \), has vertical asymptotes. Answer the following questions about these asymptotes. 1. In the interval \( -360^{\circ} \leq \alpha \leq 360^{\circ} \), at what values of \( \alpha \) are the asymptotes located? Answer: The asymptotes are at \( \alpha= \) Select

The tangent function has vertical asymptotes where the function is undefined, which occurs at odd multiples of \(90^{\circ}\). Therefore, in the interval \( -360^{\circ} \leq \alpha \leq 360^{\circ} \), the asymptotes are located at \( \alpha = -270^{\circ}, -90^{\circ}, 90^{\circ}, 270^{\circ} \). Delving a bit deeper, the behavior of the tangent function as it approaches these asymptotes is quite fascinating! As \( \alpha \) approaches \( -90^{\circ} \) or \( 90^{\circ} \), the function tends to positive or negative infinity, creating that dramatic shift. This behavior makes tangent unique among trigonometric functions and explains why it's often challenging to graph without noting asymptotes first. If you're looking to master trigonometric identities, a few classic tips can help prevent common pitfalls! Always remember that \( \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \). This will help you establish when the function is undefined (i.e., when \( \cos \alpha = 0 \)). Another common mistake is forgetting the periodic nature of the tangent function, which is periodic every \( 180^{\circ} \). Keeping this in mind will help you identify all asymptotes in wider intervals confidently!