Problem 33
Question
Find the number(s) \(x\) in the interval \([0.2 \pi]\) which satisfy the equation. $$\tan x / 2=1$$.
Step-by-Step Solution
Verified Answer
The number \(x\) in the interval [0, \(2\pi\)] that satisfies the equation \(\tan (x / 2) = 1\) is \(x=\(\pi/2\).
1Step 1: Basic Understanding
The given function is \(\tan (x / 2) = 1\). We need to find \(x\) such that its value lies in the interval [0, \(2\pi\)]. Let's start by finding when the tangent function equals 1. Note, the period of the tangent function is \(\pi\), comparing this with function we know that \(\tan^{-1}(1) = \(\pi/4\).
2Step 2: Determining the Value
Because the equation inside the tangent function is \(x/2\), we need to adjust the value to fit this. We set \(x/2 = \(\pi/4\). Solving for \(x\) gives \(x = \(\pi/2\).
3Step 3: Validating the Answer
Let's validate whether the answer meets initial criteria. The found value \(x=\(\pi/2\) is indeed in the range [0, \(2\pi\)] and thus, it is the valid solution
Other exercises in this chapter
Problem 33
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Give the domain of the function and sketch the graph. $$f(x)=2 x$$
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Indicate on a number line the numbers \(x\) that satisfy the condition. \(|x+3| \leq 0\).
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Solve the inequality and express the solution set as an interval or as the union of intervals. $$|2 x+1|
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