Problem 7 - 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. $$ \tan ^{2} \theta+3 \tan \theta+1=0 $$

Step-by-Step Solution

Answer

Solutions are \( \theta = 159^{\circ}, 339^{\circ}, 110^{\circ}, 290^{\circ} \).

Step 1: Identify the Quadratic Equation

The given equation is \( \tan^2 \theta + 3 \tan \theta + 1 = 0 \). This equation is a quadratic in terms of \( x = \tan \theta \). Therefore, we can rewrite it as \( x^2 + 3x + 1 = 0 \).

Step 2: Apply the Quadratic Formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 3 \), and \( c = 1 \). Substituting these values into the formula gives:\[x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 - 4}}{2} = \frac{-3 \pm \sqrt{5}}{2}\]

Step 3: Find Solutions for \(x = \tan \theta\)

Calculate the approximate values:\( x_1 = \frac{-3 + \sqrt{5}}{2} \approx -0.38 \) and \( x_2 = \frac{-3 - \sqrt{5}}{2} \approx -2.62 \). These are the potential values for \( \tan \theta \).

Step 4: Calculate \(\theta\) from \(\tan \theta = x\)

For \( x_1 \approx -0.38 \), \( \theta = \tan^{-1}(-0.38) \approx -20.8^{\circ} \). To fit within the interval \([0^{\circ}, 360^{\circ})\), add \( 180^{\circ} \):\( \theta \approx 159^{\circ} \).For \( x_2 \approx -2.62 \), \( \theta = \tan^{-1}(-2.62) \approx -69.7^{\circ} \). Add \( 180^{\circ} \):\( \theta \approx 110^{\circ} \).

Step 5: Validate and Adjust Solutions

Both values of \( \theta \) should be checked against the periodic nature of \( \tan \theta \), which repeats every \( 180^{\circ} \). Therefore, possible angles are:\( \theta = 159^{\circ}, 339^{\circ} \) for \( x_1 \approx -0.38 \) \( \theta = 110^{\circ}, 290^{\circ} \) for \( x_2 \approx -2.62 \).

Key Concepts

Trigonometric Equations

Trigonometric equations involve the trigonometric functions such as sine, cosine, and tangent. These equations often require solving for an unknown angle \( \theta \). When dealing with these equations, it's important to recognize the form and type of function you're working with.
The exercises like the one we have here, \( \tan^2 \theta + 3 \tan \theta + 1 = 0 \), are essentially quadratic equations in disguise, because when you replace \( \tan \theta \) with \( x \), the equation reads \( x^2 + 3x + 1 = 0 \).

• These equations usually offer multiple solutions because trigonometric functions are periodic, repeating their values in a predictable way over certain intervals.
• Hence, once you've found an angle \( \theta \) that satisfies the equation, adding or subtracting the function's period can give you additional solutions.
• Specifically for tangent, this means every solution must be adjusted within the interval using its period of \(180^{\circ}\).

Understanding trigonometric identities and relationships is crucial in solving these problems correctly.

Tangent Function

The tangent function is one of the fundamental trigonometric functions. For a given angle \( \theta \), \( \tan \theta \) represents the ratio of the sine and cosine of \( \theta \):
\[\tan \theta = \frac{\sin \theta}{\cos \theta}\]

• The tangent function is unique because it has a period of \(180^{\circ}\), meaning it repeats its pattern every \(180^{\circ}\).
• This periodicity is essential in solving tangent equations, as solutions found initially can be altered by adding \(180^{\circ}\) to find further solutions within a specified interval.
• As a function, \( \tan \theta \) is undefined where \( \cos \theta = 0 \) (at \( \theta = 90^{\circ}, 270^{\circ}, \) etc.), since division by zero is not possible.

The tangent curve spans from \(-\infty\) to \(+\infty\), rapidly increasing as \( \theta \) approaches its vertical asymptotes, the points where the function is undefined.

Angles in Degrees

Angles can be measured in various units, with degrees and radians being the most common. In trigonometry and in many practical applications, degrees are frequently used to describe angle measurements.

• One full circle is equal to \(360^{\circ}\).
• The interval from \(0^{\circ}\) to \(360^{\circ}\) is often used when identifying solutions for trigonometric equations because it conveniently covers one full revolution.
• When solving trigonometric equations, the solution may initially fall outside this range, necessitating conversions. This involves adding or subtracting \(180^{\circ}\) or \(360^{\circ}\), depending on the context, to align the results within the interval.

Converting a measurement between degrees and radians can be easily done using the conversion factor \( 180^{\circ} = \pi \text{ radians} \). This consideration is particularly important when solving equations in different units for more advanced applications.

Inverse Tangent

The inverse tangent function, noted as \( \tan^{-1} \) or \( \text{arctan} \), is used to find an angle whose tangent is a given number. Unlike the tangent, inverse tangent maps from real numbers back to angles.

• It typically returns an angle in the range \([-90^{\circ}, 90^{\circ}]\).
• This range is known as the principal range or the principal value, and it's crucial when resolving tangent equations. When an angle \( \theta \) falls outside of \( [-90^{\circ}, 90^{\circ}] \), you’ll often need to adjust it by adding \(180^{\circ}\) to position it correctly within a studied interval, such as \([0^{\circ}, 360^{\circ})\).
• This characteristic of the inverse tangent ensures that there’s always an angle \( \theta \) within its range corresponding to every real number input.

Understanding how to navigate this function and its constraints helps in successfully solving and interpreting solutions for trigonometric equations effectively.