First slide

Trigonometric equations

Question

Number of values of $x$ lying in the interval $[0, 4 π]$ and satisfying the equation $\tan 5 x + \cot 3 x = 0$ is

Moderate

A

2
B

4
C

6
D

8

Solution

$\begin{array}{l} \tan 5 x = - \cot 3 x = \tan (π / 2 + 3 x) \\ \Rightarrow 5 x = n π + \frac{π}{2} + 3 x, n \in I \\ \Rightarrow 2 x = (2 n + 1) \frac{π}{2}, n \in I \\ \Rightarrow x = (2 n + 1) \frac{π}{4}, n \in I \end{array}$
Now, $x \in [0, 4 π]$
$\begin{array}{l} \Rightarrow 0 \leq (2 n + 1) \frac{π}{4} \leq 4 π \\ \Rightarrow - \frac{1}{2} \leq n \leq \frac{15}{2} \Rightarrow 0 \leq n \leq 7 \end{array}$
$\Rightarrow$ There are 8 values of $x$ .

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