Problem 4 - Algebra 2 and Trigonometry

Question

In \(3-14\) , use the quadratic formula to find, to the nearest degree, all values of \(\theta\) in the interval \(0^{\circ} \leq \theta<360^{\circ}\) that satisfy each equation. $$ \tan ^{2} \theta-2 \tan \theta-5=0 $$

Step-by-Step Solution

Answer

\(\theta \approx 67^{\circ}, 247^{\circ}, 192^{\circ}, 12^{\circ} \).

Step 1: Identify the Quadratic Form

The given equation \( \tan^2 \theta - 2 \tan \theta - 5 = 0 \) is already in the standard quadratic form \( ax^2 + bx + c = 0 \), where \(a=1\), \(b=-2\), and \(c=-5\). The variable \(x\) in this context is \( \tan \theta \).

Step 2: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \(a=1\), \(b=-2\), and \(c=-5\) into the formula: \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-5)}}{2 \times 1}\).

Step 3: Calculate the Discriminant

Compute the discriminant: \( b^2 - 4ac = (-2)^2 - 4 \times 1 \times (-5) = 4 + 20 = 24 \). Since the discriminant is positive, there will be two real solutions for \( \tan \theta \).

Step 4: Solve for Solutions

Substitute the discriminant back into the formula: \( x = \frac{2 \pm \sqrt{24}}{2} \). Simplify \( \sqrt{24} \) to \( 2\sqrt{6} \). Thus, \( x = \frac{2 \pm 2\sqrt{6}}{2} = 1 \pm \sqrt{6} \). Thus, the two solutions for \( \tan \theta \) are \( 1 + \sqrt{6} \) and \( 1 - \sqrt{6} \).

Step 5: Find \(\theta\) for \(\tan \theta = 1 + \sqrt{6}\)

Use a calculator to determine \( \theta = \tan^{-1}(1+\sqrt{6}) \). This gives an angle \( \theta \) which should then be verified if necessary to determine the equivalent angles within \(0^{\circ}\) to \(360^{\circ}\). Recall that tangent has a period of \(180^{\circ}\).

Step 6: Find \(\theta\) for \(\tan \theta = 1 - \sqrt{6}\)

Calculate \( \theta = \tan^{-1}(1-\sqrt{6}) \) using a calculator. Verify these angles, and add \(180^{\circ}\) as needed to find equivalent solutions within the specified interval.

Step 7: Verify and Present Results

After computing the values of \(\theta\), ensure that each solution is within the interval \([0^{\circ}, 360^{\circ})\) and there are no errors in calculation. Based on calculations, the nearest degree is represented for each possible solution.

Key Concepts

Tangent Function

The tangent function, represented as \( \tan \theta \), is a fundamental trigonometric function. You might recognize it from right-angled triangles where it's defined as the ratio of the opposite side to the adjacent side. In the unit circle, tangent can also be understood as the y-coordinate divided by the x-coordinate of a point on the circle where the terminal side of an angle intersect.

• Properties: Tangent has unique properties. Unlike sine and cosine which have values between -1 and 1, tangent can take any real number values.
• Periodicity: Another key characteristic is its periodicity. Tangent repeats every \(180^{\circ}\); this means for any angle \(\theta\), \(\tan(\theta + 180^{\circ}) = \tan(\theta)\).

Understanding these properties is essential when solving trigonometric equations that involve the tangent function, especially within specified intervals like \([0^{\circ}, 360^{\circ})\).

Discriminant

The discriminant is a critical component when dealing with quadratic equations. It appears inside the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The discriminant itself is the expression below the square root: \( b^2 - 4ac \).

• Determines the Nature of Solutions: It tells you how many solutions there are and what type they will be.
• Positive Discriminant: Two distinct real solutions are present, just like in the exercise, where \( b^2 - 4ac = 24 \).
• Zero Discriminant: There is exactly one real solution.
• Negative Discriminant: There are no real solutions; instead, the solutions are complex numbers.

In our exercise, having a positive discriminant means there are two real values for \( \tan \theta \), which is pivotal for finding the angle solutions.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions like sine, cosine, or tangent. They often require solving for the angle, \( \theta \), that satisfies the equation.

• Conversion to a Known Form: Many trigonometric equations can be rearranged into a quadratic form as seen in this exercise.
• Use of Identities and Inverses: Solving these often involves using identities or inverse operations, such as \( \tan^{-1} \), to isolate the angle \( \theta \).

These equations typically have multiple solutions because of the periodic nature of trigonometric functions. For the tangent function, solutions often need to be adjusted using the function's periodicity to fit within specified intervals.

Angle Solutions

Finding angle solutions to trigonometric equations involves determining the principal and equivalent angles within a specified range. For instance, angles in the exercise need to be found within \([0^{\circ}, 360^{\circ})\).

• Principal Angle: The angle directly given by the inverse trigonometric function. For example, \( \theta = \tan^{-1}(x) \).
• Adjusting for Periodicity: Remembering that tangent has a period of \(180^{\circ}\), additional solutions are obtained by adding or subtracting \(180^{\circ}\) until all solutions fit within the given interval.

To ensure accuracy, it's important to evaluate each calculated \( \theta \) and adjust as needed to confirm it falls within the correct range. This step assures the completeness of solutions in practical trigonometric problems.