Problem 30 - Precalculus with Calculus Previews

Question

Find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \tan 4 \theta=-1 $$

Step-by-Step Solution

Answer

\( \theta = 33.75^\circ + 45^\circ n \), where \( n \in \mathbb{Z} \).

Step 1: Understanding the Problem

We need to solve for \( \theta \) in the equation \( \tan 4\theta = -1 \), where \( \theta \) is an angle measured in degrees.

Step 2: Identify the General Solutions of Tangent

Recall that the tangent function has a period of \(180^\circ\), meaning \( \tan \theta = \tan(\theta + 180^\circ n) \) for any integer \( n \). Since \( \tan 4\theta = -1 \), we need to find \( 4\theta \) such that \( \tan 4\theta = \tan(135^\circ + 180^\circ n) \).

Step 3: Set Up the Equation

Set \( 4\theta = 135^\circ + 180^\circ n \) for some integer \( n \). This step accounts for all possible angles that have a tangent of \(-1\) within the period of the tangent function.

Step 4: Solve for \( \theta \)

Divide the entire equation from Step 3 by 4 to solve for \( \theta \):\[ \theta = \frac{135^\circ + 180^\circ n}{4} \]Simplifying gives:\[ \theta = 33.75^\circ + 45^\circ n \] for \( n \in \mathbb{Z} \).

Step 5: State the General Solution

The general solution for \( \theta \) for the equation \( \tan 4 \theta = -1 \) is \( \theta = 33.75^\circ + 45^\circ n \), where \( n \) is any integer.

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan \), is a fundamental concept in trigonometry. It relates the angles in a right triangle to the ratio of the opposite side to the adjacent side. In the context of trigonometric equations, the tangent function can be a helpful tool in finding angles that satisfy a particular condition.
\( \tan \theta \) is defined as the ratio of the sine of \( \theta \) to the cosine of \( \theta \), which means:

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

This function is continuous but unlike sine and cosine, has vertical asymptotes, which occur at intervals of \( 90^\circ \).
Due to these asymptotes, the tangent function is undefined at angles where the cosine is zero. This leads us naturally to the next important concept of understanding angles in degrees.

Angle Measurement in Degrees

Angles can be measured in different units, but degrees are one of the most common. One full circle equals \( 360^\circ \), and understanding degrees is crucial in solving trigonometric equations. When discussing angles in trigonometry, the choice of degree or radian measure will affect the interpretation of the angle.
In the context of the original exercise, \( \theta \) is an angle in degrees. Degrees allow for easy visualization of angles, especially when breaking down the cycle of periodic functions like the tangent. It divides a circle into 360 parts, making it easier to grasp as many practical applications relate directly to full, half, and quarter circles.
With degrees, operations such as addition, subtraction, and multiplying by integers remain intuitive. For instance, finding the general solution for an angle using tangent arises from understanding these periodic properties in degree measurements.

Periodicity of Tangent

The tangent function exhibits periodicity, a property crucial for solving many trigonometric equations. The period of a function is the length of the interval after which the function begins to repeat. For the tangent function, this period is \(180^\circ\).
This means that if \( \tan \theta = a \), then \( \tan(\theta + 180^\circ n) = a \) for any integer \( n \). This periodicity is due to the fact that the graph of the tangent has repeating patterns every \( 180^\circ \), unlike sine and cosine which repeat every \( 360^\circ \).
This understanding is essential when deriving general solutions to trigonometric equations involving tangent, as seen in solving \( \tan 4\theta = -1 \). Recognizing the period allows for the derivation of all potential solutions without exhaustive calculation, making it a powerful aspect of solving equations involving tangent.

General Solution of Trigonometric Equations

When solving trigonometric equations, finding the general solution is a key goal. This lets you identify all possible angle values that satisfy a given equation. For tangent-related equations, understanding its periodic nature aids in deriving these solutions.
In the exercise highlighted, \( \tan 4\theta = -1 \) was solved by considering the periodicity of \(180^\circ\). By setting \( 4\theta = 135^\circ + 180^\circ n \) where \( n \) is any integer, you encapsulate all potential angles satisfying the equation.
This method essentially translates the problem into calculating shifts by the period of the tangent function to find \( \theta \). The solution \( \theta = 33.75^\circ + 45^\circ n \) takes advantage of this periodicity, ensuring that all solutions are covered. This systematic approach can be generalizable to other trigonometric equations, empowering you to confidently solve similar problems.