Ortiz Edwards
07/07/2024 · Senior High School
(2) \( \operatorname{tg} x=\frac{3}{1}, \pi
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Step-by-step Solution
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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