Problem 31
Question
Solve the equation for \(0 \leq x<2 \pi\). \(\tan (x+\pi)-\tan (\pi-x)=0\)
Step-by-Step Solution
Verified Answer
The solutions of the equation within the given range are \( x = 0, \pi \).
1Step 1: Apply Tangent Addition Properties
Firstly, recall the properties of the tangent function. We know that \(\tan (\pi + x) = \tan x\) and \(\tan (\pi - x) = -\tan x\). Applying these properties, the given equation becomes: \( \tan x + \tan x = 0 \).
2Step 2: Simplify the Equation
Simplify the equation by combining like terms, resulting in \( 2\tan x = 0 \).
3Step 3: Solve for x
Solving the equation for x, we divide both sides by 2, thus obtaining \( \tan x = 0 \). We know that the tangent function is zero at \(0\) and \( \pi\). Therefore \( x = 0, \pi \).
4Step 4: Check Solution
To ensure the solution is correct, plug the values of \( x \) back into the original equation. For both \( x = 0 \) and \( x = \pi \), the equation gives \( 0 = 0 \), thus confirming they are indeed solutions.
Key Concepts
Tangent Function PropertiesSimplification of EquationsVerification of Solutions
Tangent Function Properties
Understanding the properties of the tangent function is essential for solving trigonometric equations effectively. One fundamental property is that the tangent function is periodic with a period of \(\pi\), which means that \(\tan(x + \pi) = \tan(x)\) for any value of \(x\). This is because tangent graphs repeat every \(\pi\) radians. Another important property to note is that tangent is an odd function so it satisfies the property \(\tan(-x) = -\tan(x)\). Therefore, \(\tan(\pi - x) = -\tan(x)\).
These properties simplify trigonometric equations significantly as they allow for substitution that reduces the equation to a more manageable form. For example, in the given exercise, the equation \(\tan(x + \pi) - \tan(\pi - x) = 0\) simplifies to \(\tan x + \tan x = 0\), using the properties of the tangent function. It's essential to familiarize oneself with such properties as they are not only useful for solving equations but also for understanding the behavior of trigonometric functions in different quadrants of the unit circle.
These properties simplify trigonometric equations significantly as they allow for substitution that reduces the equation to a more manageable form. For example, in the given exercise, the equation \(\tan(x + \pi) - \tan(\pi - x) = 0\) simplifies to \(\tan x + \tan x = 0\), using the properties of the tangent function. It's essential to familiarize oneself with such properties as they are not only useful for solving equations but also for understanding the behavior of trigonometric functions in different quadrants of the unit circle.
Simplification of Equations
When it comes to algebra and trigonometry, simplification is a key process that can make complex equations more approachable. Simplifying an equation often involves combining like terms, factoring, expanding, or using trigonometric identities. It's like decluttering a room so you can better understand what's inside.
In our exercise, the process of simplification is the bridge between applying the tangent function properties and solving for \(x\). After applying the properties, we get \(2\tan x = 0\), which is a straightforward equation. Simplification has brought the problem down to its core, where the solution clearly presents itself. It is a crucial step that should be approached methodically and patiently; checking each term for common factors or identities that can make the problem more solvable by revealing a clearer path to the solution.
In our exercise, the process of simplification is the bridge between applying the tangent function properties and solving for \(x\). After applying the properties, we get \(2\tan x = 0\), which is a straightforward equation. Simplification has brought the problem down to its core, where the solution clearly presents itself. It is a crucial step that should be approached methodically and patiently; checking each term for common factors or identities that can make the problem more solvable by revealing a clearer path to the solution.
Verification of Solutions
After solving for the variable, verifying your solutions is just as important as the initial steps of solving the equation. This step confirms whether the answers obtained satisfy the original equation and are within the required domain. The process of verification involves substituting the solutions back into the original equation and checking if the equation holds true.
In our example, the solutions \(x = 0\) and \(x = \pi\) are verified by plugging them back into the original equation, \(\tan (x+\pi)-\tan (\pi-x)=0\). The verification process should not be skipped, as it can catch potential errors made in the solution process, such as arithmetic mistakes or overlooking the domain restrictions. Always remember, the proof of a solution's validity lies in its ability to withstand the scrutiny of verification.
In our example, the solutions \(x = 0\) and \(x = \pi\) are verified by plugging them back into the original equation, \(\tan (x+\pi)-\tan (\pi-x)=0\). The verification process should not be skipped, as it can catch potential errors made in the solution process, such as arithmetic mistakes or overlooking the domain restrictions. Always remember, the proof of a solution's validity lies in its ability to withstand the scrutiny of verification.