The diagram shows the graphs of $y = a \tan bx$, where $b 0$, for \(-3\pi

Question

The diagram shows the graphs of $y = a \tan bx$, where $b > 0$, for $-3\pi < x < 3\pi$ and $y = m$. The graph of $y = a \tan bx$ passes through the point $\left(\frac{\pi}{2}, 1\right)$ and intersects $y = m$ at the points $A$, $B$ and $C$. The coordinates of $C$ is $(p, m)$. Find the period of $y$ and the values of $a$ and $b$. State the coordinates of one possible point about which the graph of $y = a \tan bx$ is symmetrical and the order of rotational symmetry. Find, in terms of $p$ and $m$, the coordinates of $A$ and $B$. By sketching a proper straight line on the same diagram, find the number of solutions to the equation $k\left(x - \frac{\pi}{2}\right) + 1 - a \tan bx = 0$, where $k \le 0$, for $-3\pi < x < 3\pi$.

Accepted Answer

(i) period $= 2\pi , a = 1, b = \dfrac{1}{2}$; (ii) $(0,0)$, order $2$; (iii) $A = (p - 4\pi , m), B = (p - 2\pi , m)$; (iv) $3$ solutions

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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