Ortiz Edwards
07/07/2024 · Senior High School

(2) \( \operatorname{tg} x=\frac{3}{1}, \pi

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To solve the equation \( \tan x = 3 \) in the interval \( \pi < x < \frac{3\pi}{2} \), we first find the general solution for \( \tan x = 3 \). The tangent function has a period of \( \pi \), so the general solutions can be expressed as: \[ x = \arctan(3) + k\pi \] where \( k \) is any integer. Next, we need to find the specific solution that lies within the interval \( \pi < x < \frac{3\pi}{2} \). 1. Calculate \( \arctan(3) \): - The value of \( \arctan(3) \) is approximately \( 1.249 \) radians. 2. Now, we find the first solution in the specified interval: - For \( k = 1 \): \[ x = \arctan(3) + \pi \approx 1.249 + 3.142 \approx 4.391 \text{ radians} \] 3. Check if this value is within the interval \( \pi < x < \frac{3\pi}{2} \): - \( \pi \approx 3.142 \) - \( \frac{3\pi}{2} \approx 4.712 \) Since \( 4.391 \) is indeed between \( 3.142 \) and \( 4.712 \), we conclude that the solution to the equation \( \tan x = 3 \) in the interval \( \pi < x < \frac{3\pi}{2} \) is: \[ x \approx 4.391 \] Thus, the final answer is: \[ x = \arctan(3) + \pi \]

Quick Answer

The solution to \( \tan x = 3 \) in the interval \( \pi < x < \frac{3\pi}{2} \) is \( x = \arctan(3) + \pi \).
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