# Vertical Asymptotes of Trigonometric Functions Let’s find the expressions for the **vertical asymptotes** for each function. --- ## (a) \( f(x) = 2\sec(x + \frac{\pi}{6}) - 1 \) ### **Step 1:** Recall that \(\sec(\theta)\) has vertical asymptotes where \(\cos(\theta) = \). ### **Step 2:** Set the inside of cosine to zero: \[ \cos(x + \frac{\pi}{6}) = \] ### **Step 3:** The cosine function is zero at: \[ x + \frac{\pi}{6} = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} \] ### **Step 4:** Solve for \(x\): \[ x = \frac{\pi}{2} - \frac{\pi}{6} + k\pi = \frac{3\pi - \pi}{6} + k\pi = \frac{2\pi}{6} + k\pi = \frac{\pi}{3} + k\pi \] ### **Final Answer (a):** \[ \boxed{x = \frac{\pi}{3} + k\pi, \quad k \in \mathbb{Z}} \] --- ## (b) \( g(x) = \cot(4(x - \frac{\pi}{3})) \) ### **Step 1:** Recall that \(\cot(\theta)\) has vertical asymptotes where \(\sin(\theta) = \). ### **Step 2:** Set the inside of sine to zero: \[ \sin(4(x - \frac{\pi}{3})) = \] ### **Step 3:** Sine is zero at integer multiples of \(\pi\): \[ 4(x - \frac{\pi}{3}) = k\pi, \quad k \in \mathbb{Z} \] ### **Step 4:** Solve for \(x\): \[ x - \frac{\pi}{3} = \frac{k\pi}{4} \implies x = \frac{\pi}{3} + \frac{k\pi}{4} \] ### **Final Answer (b):** \[ \boxed{x = \frac{\pi}{3} + \frac{k\pi}{4}, \quad k \in \bb{Z}} \] --- ## (c) \( f(x) = -3\csc(\frac{}{2} x - \frac{\pi}{2}) + 2 \) ### **Step 1:** Recall that \(\csc(\theta)\) has vertical asymptotes where \(\sin(\theta) = \). ### **Step 2:** Set the argument to zeros of sine: \[ \sin\left(\frac{1}{2}x - \frac{\pi}{2}\right) = \] ### **Step 3:** \[ \frac{1}{2}x - \frac{\pi}{2} = k\pi, \quad k \in \mathbb{Z} \] ### **Step 4:** Solve for \(x\): \[ \frac{1}{2}x = k\pi + \frac{\pi}{2} \] \[ x = 2k\pi + \pi \] ### **Final Answer (c):** \[ \boxed{x = \pi + 2k\pi, \quad k \in \mathbb{Z}} \] --- ## **Summary Table** | Function | Vertical Asymptotes | |-----------------------------------------------------|-----------------------------| | \( 2\sec(x+\frac{\pi}{6})-1 \) | \( x = \frac{\pi}{3} + k\pi \) | | \( \cot(4(x-\frac{\pi}{3})) \) | \( x = \frac{\pi}{3} + \frac{k\pi}{4} \)| | \( -3\csc(\frac{1}{2}x-\frac{\pi}{2})+2 \) | \( x = \pi + 2k\pi \) |