Problem 55 - Algebra and Trigonometry

Question

In Exercises \(55-58,\) use a graph to solve each equation for \(-2 \pi \leq x \leq 2 \pi\) $$ \tan x=-1 $$

Step-by-Step Solution

Answer

The solutions of the given equation \(\tan(x) = -1\) in the range \(-2\pi \leq x \leq 2\pi\) are \(x = 3\pi/4\), \(x = -\pi/4\), and \(x = 7\pi/4\).

Step 1: Use the Basic Property of Tangent function

Tan function is negative in the second and fourth quadrants (considering the rule ASTC or 'All Students Take Calculus' where A is for All and refers to the first quadrant where all trigonometric functions are positive, S is for sine and refers to the second quadrant where sine is positive, T is for tangent and refers to the third quadrant where tangent is positive, and C is for cosine and refers to the fourth quadrant where cosine is positive). Now, recall that tan(θ) = -1 only at angles θ = \(-\pi/4\) and \(3\pi/4\). So, we get the solutions for the first period of the tangent function.

Step 2: Add Periods of Tangent Function

The tangent function has a period of \(\pi\), so we can find the other solutions in the given range by adding and subtracting \(\pi\) to the obtained solutions from the first step. So for the solutions in the second period, we add \(\pi\) to both the obtained solutions from Step 1 which gives us angles \(3\pi/4 + \pi = 7\pi/4\) and \(-\pi/4 + \pi = 3\pi/4\). As \(7\pi/4\) is more than \(2\pi\), it is not in the specified range, so we ignore it. Therefore, we have three solutions that satisfy the given condition within the specified domain.

Step 3: Provide Final Solution

The solutions for the equation \(\tan(x) = -1\) in the domain \(-2\pi \leq x \leq 2\pi\), are \(x = 3\pi/4\), \(x = 7\pi/4\) and \(x = -\pi/4\).

Key Concepts

Graphical Solutions

Graphical solutions provide a visual interpretation of mathematical problems, turning complex equations into intuitive pictorial representations. When you're solving trigonometric equations like \(\tan(x) = -1\), using a graph becomes powerful. Here's why:

• Graphs allow you to observe periodic behavior, which is essential for trigonometric functions.
• You can visualize where the function intersects a horizontal line (representing the solution, in this case, \(y = -1\)).
• Using the graph, you can easily identify all potential solutions in a specified interval.

The tangent function graph is unique due to its repeated cycles or periods. Each period of the tangent curve looks identical, which helps when identifying solutions over multiple intervals. So, by plotting the graph of the tangent function from \(-2\pi\) to \(2\pi\), you can spot where the function equals \(-1\), and identify solutions without diving into complicated algebra.

Remember, in trigonometry, a single graphical observation can often simplify what would otherwise be a cumbersome calculation. It's a visual shortcut that leads to easy and quick insights.

Tangent Function

The tangent function stands out from other trigonometric functions due to its distinct properties. Let's explore why this function is particularly special:

• Periodicity: While the sine and cosine functions have a period of \(2\pi\), the tangent function has a period of \(\pi\). This means that \(\tan(x + \pi) = \tan(x)\), making it easier to find multiple solutions across intervals.
• Behavior in Quadrants: The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants. Using the ASTC rule, you know exactly where to look for negative or positive values of tangent.
• Asymptotic Nature: Unlike other trigonometric functions, tangent has asymptotes at every odd multiple of \(\pi/2\), where it's undefined. This leads to its characteristic wave pattern on graphs.

To solve equations involving the tangent function, these attributes are vital. For example, knowing where the function is positive or negative aids in quickly identifying solution intervals. Additionally, knowing the period allows you to generate other solutions by simply adding or subtracting \(\pi\). With these tips, you can master the behavior of the tangent function.

Trigonometric Identities

Trigonometric identities are critical tools in simplifying equations and finding solutions. They relate the angles and functions to one another, providing shortcuts and transforming complex problems into more manageable ones.

Here are some essential trigonometric identities that can assist greatly in solving equations like \(\tan(x) = -1\):

• Basic Identities: These include \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), which helps in transforming the tangent function equation into sine and cosine functions when needed.
• Reciprocal Identities: Though not directly related to tangent, identities like \(\cot(x) = \frac{1}{\tan(x)}\) could be useful when analyzing equations where tangent can be expressed in another form.
• Pythagorean Identities: For instance, \(1 + \tan^2(x) = \sec^2(x)\), which might help in more advanced transformations or verification of solutions.

By understanding and applying these identities, you can simplify trigonometric equations significantly. This allows you to focus on finding the fundamental solutions and then extends them across the desired interval, much like in our exercise scenario. Remember, identities are not just equations—they are powerful tools to make problems easier to solve.