Problem 18 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ 4 \tan \theta=3+\tan \theta $$

Step-by-Step Solution

Answer

The solutions for the equation \(4 \tan \theta = 3 + \tan \theta\) within the range \(0 \leq \theta < 2\pi\) are \(\theta = \pi/4\) and \(\theta = 5\pi/4\).

Step 1: Simplification

Firstly, rearrange the equation in order to bring all terms to one side. This will give you \\(4 \tan \theta - \tan \theta - 3 = 0\), which simplifies to \(3 \tan \theta - 3 = 0\).

Step 2: Solve for \(\tan \theta\)

Next, solve for \(\tan \theta\) to find \(\tan \theta = 1\).

Step 3: General Solutions

Considering the periodic nature of the trigonometric functions, the general solutions can be found using the formula \(\theta = n\pi + \tan^{-1}(a)\), where n is any integer and a is the value obtained in the previous step. In this case, \(\theta = n\pi + \tan^{-1}(1)\) gives the general solutions.

Step 4: Specific Solutions

Since the question asks for solutions in the range \(0 \leq \theta < 2\pi\), you should substitute integer values of n into the general solution such that \(\theta\) falls within this range. When \(n = 0\) or \(n = 1\), you get \(\theta = \pi/4\) or \(\theta = 5\pi/4\) respectively.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics as they relate angles to ratios. These functions include sine, cosine, and tangent, among others. They are widely used in various fields such as physics, engineering, and even in daily life activities like navigation. Understanding these functions is key to solving many mathematical problems, especially those involving angles and periodic behavior.
In the exercise presented, the focus is on the tangent function, one of the fundamental trigonometric functions. The tangent of an angle \(\theta\) is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. It is often denoted as \(\tan \theta\).

• The range of the tangent function is all real numbers.
• The function is periodic with a period of \(\pi\).
• Trigonometric functions like tangent are used to model waves and oscillations due to their periodic nature.

General Solutions

Finding the general solution to a trigonometric equation involves using the periodic properties of the trigonometric functions. The key here is to determine how changes in the angle affect the function systematically and repeatedly.
In most trigonometric equations, the solutions occur periodically. For the tangent function, if \(\theta\) is a solution, then any angle of the form \((\theta + n\pi)\), where \(n\) is an integer, is also a solution. This characteristic is because the tangent function repeats every \(\pi\) radians.
For our exercise, we identified the general solution pattern for the equation \(\tan \theta = 1\). With this, if \(\theta = \pi/4\)is a solution, then any angle \(\theta = n\pi + \pi/4\) is a general solution. This concept allows us to list all possible angles that satisfy the equation, considering the infinite cyclical nature of trigonometric functions.

Specific Solutions

When dealing with trigonometric equations, sometimes we need to find solutions within a specific interval. This is where the concept of specific solutions comes in, especially when a range is given, such as from \(0\) to \(2\pi\).
In the given exercise, once the general solutions were found, the task was to identify which of these solutions fell within the specified range. By examining each general solution, we make sure that the angle \(\theta\) satisfies the equation while staying within this range.

• For \(n = 0\), the specific solution is \(\theta = \pi/4\).
• For \(n = 1\), the specific solution is \(\theta = 5\pi/4\).

Thus, these specific solutions indicate the points where the tangent reaches the desired value within the interval \([0, 2\pi)\).

Tangent Function

The tangent function, denoted as \(\tan\theta\), is one of the primary trigonometric functions. It plays a significant role in various mathematical applications due to its unique properties. Unlike sine and cosine, the tangent function can take any real number as a value, thanks to its asymptotes.
The function is periodic, repeating itself every \(\pi\) radians, which means it has the same value for angles that differ by \(\pi\). The tangent function is especially important when dealing with right-angled triangles or when analyzing periodic phenomena like sound waves or light waves.

• The tangent of \(\theta\) is zero whenever \(\theta\) is a multiple of \(\pi\).
• It has vertical asymptotes, or lines it cannot touch or cross, at \(\theta = \pi/2 + n\pi\).
• Solving equations with the tangent function often involves considering these traits, particularly the infinite nature of potential solutions.