Question

trigonometric ratios and functions exam
c) tan(39°)=15/11
d) tan(39°)=11/15
question 14 (5 points)
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which two equations are the equations of vertical asymptotes of the function y = 5/3 tan(3/4x)?
a) x = 4π/3 and x = -4π/3
b) x = 2π/3 and x = -2π/3
c) x = 0 and x = 4π/3
d) x = 0 and x = 2π/3

Explanation:

Step1: Recall tangent - asymptote formula

The vertical asymptotes of the tangent function $y = A\tan(Bx)$ occur at $Bx=\frac{\pi}{2}+k\pi$, where $k\in\mathbb{Z}$. For the function $y = \frac{5}{3}\tan(\frac{3}{4}x)$, we set $\frac{3}{4}x=\frac{\pi}{2}+k\pi$.

Step2: Solve for $x$

Multiply both sides of the equation $\frac{3}{4}x=\frac{\pi}{2}+k\pi$ by $\frac{4}{3}$. We get $x=\frac{4}{3}(\frac{\pi}{2}+k\pi)=\frac{2\pi}{3}+\frac{4k\pi}{3}$.

Step3: Find two asymptotes

When $k = 0$, $x=\frac{2\pi}{3}$. When $k=- 1$, $x=\frac{2\pi}{3}-\frac{4\pi}{3}=-\frac{2\pi}{3}$.

Answer:

B. $x = \frac{2\pi}{3}$ and $x=-\frac{2\pi}{3}$