Problem 65 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ \tan ^{2} \theta+3 \tan \theta+2=0 $$

Step-by-Step Solution

Answer

The approximate solutions are \( \theta = 120^\circ, 135^\circ, 300^\circ, 315^\circ \).

Step 1: Identify the type of equation

The given equation is a quadratic equation in terms of \( \tan \theta \). It is expressed as \( \tan^2 \theta + 3 \tan \theta + 2 = 0 \).

Step 2: Solve the quadratic equation

To solve for \( \tan \theta \), we recognize that this is a standard quadratic equation in the form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 3 \), and \( c = 2 \). We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substituting, we get: \[ \tan \theta = \frac{-3 \pm \sqrt{3^2 - 4 \times 1 \times 2}}{2 \times 1} \].

Step 3: Calculate discriminant and roots

Calculate the discriminant: \[ 3^2 - 4 \cdot 1 \cdot 2 = 9 - 8 = 1 \]. Since the discriminant is positive, there are two distinct real roots: \[ \tan \theta = \frac{-3 + 1}{2} = -1 \] and \[ \tan \theta = \frac{-3 - 1}{2} = -2 \].

Step 4: Solve for angles within the interval

Find the angles \( \theta \) where \( \tan \theta = -1 \) and \( \tan \theta = -2 \) within \([0^\circ, 360^\circ)\). For \( \tan \theta = -1 \), \( \theta = 135^\circ \) and \( 315^\circ \). For \( \tan \theta = -2 \), we approximate to nearest 10\(^\circ\), \( \theta \approx 120^\circ \) and \( 300^\circ \).

Step 5: Verify results for consistency

Review the interval and angles obtained: All calculated angles \( \theta = 135^\circ, 315^\circ, 120^\circ, 300^\circ \) are within \([0^\circ, 360^\circ)\) and accurate to the nearest 10\(^\circ\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equations where the highest degree is 2. These equations take the general form: \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic equations can be solved using various methods, with the quadratic formula being one of the most common approaches. It is given by:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides solutions for the variable \( x \), known as "roots" of the equation. In the context of trigonometric equations like \( \tan^2 \theta + 3 \tan \theta + 2 = 0 \), "tan \( \theta \)" acts as the variable "x", transforming the trigonometric equation into a quadratic equation. Thus, understanding quadratic equations equips you with the tools to tackle various mathematical problems, including those involving trigonometric functions.

Tangent Function

The tangent function, \( \tan \theta \), is one of the primary trigonometric functions. It relates the angle \( \theta \) in a right triangle to the ratio of the length of the opposite side to the adjacent side via the equation:

• \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

The tangent function is periodic, repeating every \( 180^\circ \). This periodicity is vital when finding all possible angles that satisfy a given equation, such as when solving \( \tan^2 \theta + 3 \tan \theta + 2 = 0 \). The solutions to such equations must consider this periodic behavior, particularly when the angles must lie within a specific range, such as \( [0^\circ, 360^\circ) \). It is important to note how, at particular angles such as \( 135^\circ \) and \( 315^\circ \), the tangent function yields specific results, indicating symmetry and periodic repetition of values. This concept helps to determine the full set of solutions for trigonometric equations.

Discriminant

The discriminant in a quadratic equation \( ax^2 + bx + c = 0 \) is expressed as \( \Delta = b^2 - 4ac \). This component of the quadratic formula reveals critical information about the nature of the roots of the equation:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, also called a "repeated" root.
• If \( \Delta < 0 \), there are no real roots, only complex roots.

In the provided solution, the discriminant calculation of \( 3^2 - 4 \times 1 \times 2 = 1 \) which is greater than zero, indicates two distinct real roots for the equation in terms of \( \tan \theta \). Understanding the discriminant is crucial for predicting the number and type of solutions, which is particularly significant in trigonometric equations where the roots represent potential angles.