Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the multiple-angle equation. $$\tan 2 x=-1$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{3\pi}{8} + \frac{k\pi}{2}\) and \(x = \frac{7\pi}{8} + \frac{k\pi}{2}\) where \(k\) is an integer.

Step 1: Determine where tangent equals -1

First, it is necessary to find those angles in the basic period \([0, \pi)\) where \(\tan(\theta) = -1\). Note that these are \(\frac{3\pi}{4}\) and \(\frac{7\pi}{4}\)

Step 2: Substitute into given equation

Next, substitute back into the given equation. We know that \(\tan(2x) = -1\), so we can solve \(2x = \frac{3\pi}{4} + k\pi\) and \(2x = \frac{7\pi}{4} + k\pi\) where \(k\) is an integer.

Step 3: Solve for x

Finally, solving the above equation for \(x\) gives \(x = \frac{3\pi}{8} + \frac{k\pi}{2}\) and \(x = \frac{7\pi}{8} + \frac{k\pi}{2}\), where \(k\) is an integer.

Key Concepts

Tangent Function

The tangent function, denoted as \(\tan\), is one of the fundamental trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, but when dealing with the unit circle, it represents the ratio of the y-coordinate to the x-coordinate of a point on the unit circle.

\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)

When exploring the properties of the tangent function, it's crucial to remember that it is periodic with a period of \(\pi\), which means that \(\tan(\theta + k\pi) = \tan(\theta)\) for any integer \(k\). This periodic nature causes the function to have an infinite number of angles where it takes on the same value. Another important property is that the tangent function has asymptotes where the cosine function equals zero, specifically at \(\frac{\pi}{2} + k\pi\).

Understanding the behavior of the tangent function is essential when solving trigonometric equations involving it, as you need to consider its periodicity, the quadrants in which it is positive or negative, and where the function is undefined.

Solving Trigonometric Equations

Solving trigonometric equations involves finding all the angles that satisfy an equation involving trigonometric functions. This process is often based on the unit circle and the properties of trigonometric functions.

Follow these steps for solving a typical trigonometric equation:

• Determine the basic solutions within a fundamental period, like \([0, 2\pi)\) or \([0, \pi)\) for the tangent function.
• Use the function's periodicity to write the general solution that accounts for all possible angles that satisfy the equation.
• When there is a multiple-angle involved, like in our example \(\tan 2x = -1\), you need to solve for the multiple-angle first and then divide to get the variable alone.
• Apply any given restrictions, such as bounds for the variable, to find the specific set of solutions within that restriction.

Always remember to check if the solutions lie within the domain of the trigonometric function and the constraints of the particular problem you're working on.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variable where the functions are defined. These identities allow us to simplify and manipulate trigonometric expressions and solve trigonometric equations. Some of the most commonly used identities include Pythagorean identities, angle addition formulas, and double angle formulas.

Here are a couple of fundamental identities:

• Pythagorean Identity: \(\sin^2(\theta) + \cos^2(\theta) = 1\)
• Double Angle Formulas for Tangent: \(\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}\)

The utilization of these identities can greatly simplify the process of solving trigonometric equations. For example, recognizing when a double angle formula can be used might turn a complex equation into one that is more straightforward to solve. Hence, being familiar with these identities and knowing when to apply them is a powerful tool in trigonometry.