Problem 34
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
Key Concepts
Tangent FunctionPeriodicityInverse Trigonometric FunctionSolution Validation
Tangent Function
The tangent function, denoted as \( \tan(x) \), is a fundamental trigonometric function. It relates to the sine and cosine functions through the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). This function has a characteristic behavior where it repeats, or cycles, at regular intervals known as its period. In the context of solving equations, knowing the values where \( \tan(x) \) takes specific numbers, such as \( 1 \), is crucial. For instance, \( \tan(\pi/4) = 1 \) is a key value that often appears in problems involving the tangent function.
Understanding this value allows us to solve equations by determining the angle that makes the tangent function equal \( 1 \). This understanding is the first step in systematically solving trigonometric equations that involve the tangent function.
Understanding this value allows us to solve equations by determining the angle that makes the tangent function equal \( 1 \). This understanding is the first step in systematically solving trigonometric equations that involve the tangent function.
Periodicity
Periodicity in trigonometry refers to the repeating pattern of trigonometric functions. The tangent function, specifically, has a period of \( \pi \). This means that \( \tan(x) = \tan(x + n\pi) \) for any integer \( n \).
When solving equations like \( \tan(x/2) = 1 \), understanding periodicity helps us know that solutions can repeat at regular intervals. In practical terms, this implies that once a solution is found, another solution can be determined by adding multiples of the function's period to the initial solution.
This concept becomes especially useful when considering intervals, as it ensures all possible solutions are accounted for within a given range.
When solving equations like \( \tan(x/2) = 1 \), understanding periodicity helps us know that solutions can repeat at regular intervals. In practical terms, this implies that once a solution is found, another solution can be determined by adding multiples of the function's period to the initial solution.
This concept becomes especially useful when considering intervals, as it ensures all possible solutions are accounted for within a given range.
Inverse Trigonometric Function
Inverse trigonometric functions help us find angles given specific trigonometric ratios. For tangent, the inverse function is \( \tan^{-1}(x) \) or \( \arctan(x) \). It gives us the angle whose tangent is \( x \).
In solving \( \tan(x/2) = 1 \), we use the inverse function to find that \( x/2 = \tan^{-1}(1) = \pi/4 \). This step is crucial as it provides the specific angle value necessary to solve the original equation. From here, algebraic manipulation helps us to find the desired angle \( x \).
In solving \( \tan(x/2) = 1 \), we use the inverse function to find that \( x/2 = \tan^{-1}(1) = \pi/4 \). This step is crucial as it provides the specific angle value necessary to solve the original equation. From here, algebraic manipulation helps us to find the desired angle \( x \).
- Use the inverse function to find angle values.
- Apply algebra to solve for the variable.
- Consider the function's behavior in its specific interval.
Solution Validation
Solution validation ensures that the solution meets all conditions of the problem, including specified intervals. In this case, after finding \( x = \pi/2 \), it's essential to verify that this value lies within the original interval \([0, 2\pi]\).
Validation confirms that the solution is not only mathematically correct but also practical within the given constraints. It helps avoid potential errors that might arise from overlooking the interval limits or the periodic nature of trigonometric functions.
Validation confirms that the solution is not only mathematically correct but also practical within the given constraints. It helps avoid potential errors that might arise from overlooking the interval limits or the periodic nature of trigonometric functions.
- Check solutions against the stated interval.
- Verify correctness by substituting back into the original equation.
- Ensure that all criteria of the problem are satisfied.
Other exercises in this chapter
Problem 34
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Sketch the set on a number line. \([3, \infty)\).
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