Problem 30 - Algebra 2

Question

Solve each equation for \(0 \leq \theta<2 \pi\). $$ \tan \theta=\tan ^{2} \theta $$

Step-by-Step Solution

Answer

The solutions to the equation \(\tan \theta = \tan^2 \theta\) are \(\theta = 0\), \(\pi\), \(\pi/4\), and \(5 \pi/4\).

Step 1: Rearrange the equation

Rearrange the equation in quadratic form: \(\tan^2 \theta - \tan \theta = 0\) Then, factor out \(\tan \theta\): \(\tan \theta (\tan \theta - 1) = 0\). This gives us two equations to solve: \(\tan \theta = 0\) and \(\tan \theta = 1\).

Step 2: Solve first equation

Starting with \(\tan \theta = 0\): The tangent of an angle is 0 at \(\theta = 0\) and \(\theta = \pi\) within the given range.

Step 3: Solve second equation

Following with \(\tan \theta = 1\): The tangent of an angle is 1 at \(\theta = \pi/4\) and \(\theta = 5\pi/4\) within the given range.

Step 4: Gather all solutions

Gather all the solutions from steps 2 and 3. The values for \(\theta\) that satisfy the equation are \(\theta = 0\), \(\pi\), \(\pi/4\), and \(5 \pi/4\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a key concept in algebra and trigonometry. A quadratic equation is any equation that can be rearranged in standard form as \(ax^2 + bx + c = 0\).
Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\). These kinds of equations often appear in various problems across mathematics.

In the provided exercise, the equation \(\tan \theta = \tan^2 \theta\) was transformed into a quadratic equation form: \(\tan^2 \theta - \tan \theta = 0\). This allows us to factor and solve the equation similarly to other types of quadratic equations.

• Quadratic equations may involve factoring to find solutions.
• The solutions can be found using algebraic methods or the quadratic formula.

The versatility of quadratic equations makes them essential in solving trigonometric exercises. The particular structure of the original problem underscores the idea that understanding quadratic equations is crucial for solving these tasks effectively.

Exploring the Tangent Function

The tangent function, denoted as \(\tan \theta\), is one of the six fundamental trigonometric functions. It relates to the angles and lengths of right triangles and is defined as the ratio of the sine to the cosine function: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

For angles where \(0 \leq \theta < 2\pi\), the tangent function assumes a distinct behavior as it cycles through its period of \(\pi\). This periodic nature means that \(\tan \theta\) returns to a value of zero at \(\theta = 0\) and \(\theta = \pi\).

• The tangent function is undefined where \(\cos \theta = 0\).
• It has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\).

In the original problem, solving \(\tan \theta = 0\) gives us solutions \(\theta = 0\) and \(\theta = \pi\), while \(\tan \theta = 1\) provides the solutions \(\theta = \frac{\pi}{4}\) and \(\theta = \frac{5\pi}{4}\). Understanding the tangent function's properties is instrumental in efficiently solving such trigonometric equations.

Factoring Equations for Solutions

Factoring is a powerful technique used to simplify equations and find solutions. It involves expressing a polynomial as a product of simpler polynomials. Factoring is particularly useful with quadratic equations, where the solutions can be directly obtained by setting each factor equal to zero.

In our exercise, the equation \(\tan \theta(\tan \theta - 1) = 0\) results from factoring the quadratic equation \(\tan^2 \theta - \tan \theta = 0\). This reveals two possible scenarios: either \(\tan \theta = 0\) or \(\tan \theta = 1\).

• Factoring can reduce complex equations to simpler, solvable forms.
• It helps in identifying key values that satisfy the equation.

Despite being a straightforward technique, factoring requires practice to recognize and apply in various mathematical contexts. Factoring not only streamlines the solution process but also enhances one’s problem-solving skills by breaking down challenging mathematical problems into manageable parts.