Problem 16 - Algebra and Trigonometry

Question

\(3-16 \cdot\) Solve the given equation. $$ 2 \tan \theta+\sec ^{2} \theta=4 $$

Step-by-Step Solution

Answer

\(\theta = n\pi + \frac{\pi}{4}\) or \(\theta = n\pi + \tan^{-1}(-3)\), where \(n\) is an integer.

Step 1: Understand the Equation

We are given the trigonometric equation: \(2 \tan \theta + \sec^2 \theta = 4\). Our goal is to solve for \(\theta\). It's helpful to recall that \(\sec^2 \theta = 1 + \tan^2 \theta\). This identity will be useful in simplifying the equation.

Step 2: Substitute Identity

Replace \(\sec^2 \theta\) in the original equation with \(1 + \tan^2 \theta\). This gives us: \(2 \tan \theta + 1 + \tan^2 \theta = 4\).

Step 3: Simplify the Equation

Rearrange and simplify the equation: \(\tan^2 \theta + 2 \tan \theta + 1 = 4\). This simplifies to \(\tan^2 \theta + 2 \tan \theta + 1 - 4 = 0\), which is \(\tan^2 \theta + 2 \tan \theta - 3 = 0\).

Step 4: Solve the Quadratic Equation

Now solve the quadratic equation \(x^2 + 2x - 3 = 0\), where \(x = \tan \theta\). Factor the quadratic: \((x + 3)(x - 1) = 0\). This gives the solutions \(x = -3\) and \(x = 1\).

Step 5: Solve for \(\theta\)

1. If \(\tan \theta = -3\), find \(\theta\) using the tangent inverse function. However, common angles often do not correspond to this value, but you can express the result in terms of \(n\pi + \tan^{-1}(-3)\).2. If \(\tan \theta = 1\), solve for \(\theta\). The basic angle is \(\theta = \frac{\pi}{4}\), and since the tangent function is periodic with period \(\pi\), general solutions are \(\theta = n\pi + \frac{\pi}{4}\), where \(n\) is an integer.

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan \theta \), is a fundamental trigonometric function. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The tangent function is periodic with a period of \( \pi \) and has a vertical asymptote wherever the cosine function is zero. This means:

• \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)
• It cycles every \( \pi \) radians, leading to repetitive pattern after this interval.
• Values of \( \theta \) that result in a zero cosine value make \( \tan \theta \) undefined as these are vertical asymptotes.

For solving equations involving the tangent function, we often look for specific angle solutions like \( \theta = \tan^{-1}(x) \), where \( x \) is the value of tan. Be aware that tangent can take any real value.

Secant Function

The secant function, expressed as \( \sec \theta \), provides another view of angle relations. It is defined as the reciprocal of the cosine function. In mathematical terms:

• \( \sec \theta = \frac{1}{\cos \theta} \)
• This implies that \( \sec^2 \theta = 1 + \tan^2 \theta \), a crucial identity used to transform trigonometric expressions.
• It is undefined where cosine equals zero, similar to tangent's conditions.

Understanding secant in the context of equations provides a powerful approach, especially when dealing with transformations between different trigonometric forms and their identities.

Quadratic Equation

Quadratic equations appear frequently in math, including trigonometry. They are of the form \( ax^2 + bx + c = 0 \). Solving these equations is crucial when \( x \) is replaced with a trigonometric function like \( \tan \theta \).

• Our original equation simplifies to \( \tan^2 \theta + 2 \tan \theta - 3 = 0 \).
• To solve, factor the equation: \((x + 3)(x - 1) = 0\).
• This means \( \tan \theta = -3 \) or \( \tan \theta = 1 \).

Solutions are not just numbers; they have trigonometric implications. Recognize when you are working with trigonometric identities to find solutions accurately.

Angle Solutions

Finding angle solutions from trigonometric equations involves recognizing which angles satisfy the functions. Specifically for the tangent function, the general solution is defined by its periodic nature:

• If \( \tan \theta = x \), then \( \theta = n\pi + \tan^{-1}(x) \), where \( n \) is an integer encompassing all period shifts.
• For \( \tan \theta = 1 \), the primary angle is \( \theta = \frac{\pi}{4} \). Solutions leverage periodicity: \( \theta = n\pi + \frac{\pi}{4} \).
• For \( \tan \theta = -3 \), the angle is less typical, expressed as \( \theta = n\pi + \tan^{-1}(-3) \).

These formulas help find solutions across the entire range of possible angles in trigonometric contexts.