Problem 65 - Precalculus: Functions and Graphs

Question

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 solutions are \(120^\circ, 135^\circ, 300^\circ, \) and \( 315^\circ\).

Step 1: Identify the Quadratic Form

The given equation \( \tan^2 \theta + 3 \tan \theta + 2 = 0 \) is a quadratic equation in terms of \( \tan \theta \). We can solve this as if \( \tan \theta \) were a variable, say \( x \). Thus, rewrite the equation as \( x^2 + 3x + 2 = 0 \).

Step 2: Factor the Quadratic Equation

To find the roots of the equation \( x^2 + 3x + 2 = 0 \), factor it to obtain \( (x + 1)(x + 2) = 0 \).

Step 3: Solve for \( x \)

Set each factor equal to zero: \( x + 1 = 0 \) and \( x + 2 = 0 \). Solving these gives \( x = -1 \) and \( x = -2 \). Thus, the solutions for \( \tan \theta \) are \( \tan \theta = -1 \) and \( \tan \theta = -2 \).

Step 4: Solve for \( \theta \) when \( \tan \theta = -1 \)

For \( \tan \theta = -1 \), \( \theta = 135^\circ \) and \( \theta = 315^\circ \) are solutions because tangent is negative in the second and fourth quadrants.

Step 5: Solve for \( \theta \) when \( \tan \theta = -2 \)

For \( \tan \theta = -2 \), the solutions need to be calculated using a calculator. Find the reference angle for positive \( \tan^{-1}(2) \) which is approximately \( 63.4^\circ \). Since tangent is negative in the second and fourth quadrants, the angles are \( 180^\circ - 63.4^\circ \approx 116.6^\circ \) and \( 360^\circ - 63.4^\circ \approx 296.6^\circ \). Round these to the nearest 10 degrees: \( 120^\circ \) and \( 300^\circ \).

Step 6: List All Solutions in the Interval

Collecting all unique solutions that fall within the interval \([0^\circ, 360^\circ)\), we have \( \theta = 120^\circ, 135^\circ, 300^\circ, \) and \( 315^\circ \), rounded to the nearest 10 degrees.

Key Concepts

Tangent Function

The tangent function is one of the primary trigonometric functions, just like sine and cosine. It is defined as the ratio of the sine function to the cosine function. Formulaically, it is expressed as \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

The tangent function is periodic with a period of \(180^\circ\) or \(\pi\) radians. This means that the function repeats its values every \(180^\circ\).

Important aspects of the tangent function include:

• Undefined values at \(90^\circ + k \cdot 180^\circ\) for integer \(k\), because cosine is zero at these angles.
• Its range includes all real numbers, differing from sine and cosine, which are bounded between \(-1\) and \(1\).
• The tangent function is negative in the second and fourth quadrants, which is crucial for solving equations like the one described where you need \(\tan \theta = -1\) or \(-2\).

Understanding the properties of tangent helps in finding solutions in different quadrants based on the sign and value of the tangent function.

Quadratic Equation

A quadratic equation is a second-degree polynomial represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is an unknown variable. The equation \(\tan^2 \theta + 3 \tan \theta + 2 = 0\) can be treated as a quadratic equation by considering \(\tan \theta\) as a variable. This reformulation transforms the equation into \(x^2 + 3x + 2 = 0\) where \(x = \tan \theta\).

Quadratic equations can be solved through various methods including:

• Factoring: This involves expressing the quadratic as a product of its linear factors, which is done here to get \((x+1)(x+2) = 0\).
• Quadratic Formula: If the equation doesn't factor easily, it can be solved using the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Completing the Square: Another method less commonly used but essential in higher algebraic contexts.

By solving the factors \((x + 1)\) and \((x + 2)\), we find the solutions for \(x\) as \(-1\) and \(-2\), which directly gives the potential values for \(\tan \theta\). Tackling these values leads to identifying suitable angle measures \(\theta\).

Reference Angle

A reference angle is a crucial concept in trigonometry which helps in finding angles across different quadrants. It is the smallest angle that a standard position angle makes with the x-axis.

To find a reference angle when dealing with the tangent being negative, it is helpful to first find the angle with the positive tangent value. For instance, for \(\tan \theta = -2\), calculate \(\tan^{-1}(2)\). This gives an angle from standard position where tangent is positive, approximately \(63.4^\circ\) in this instance.

To convert this reference angle to the required angle in the second and fourth quadrants (where tangent is negative):

• Fourth Quadrant: Subtract from \(360^\circ\), e.g., \(360^\circ - 63.4^\circ \approx 296.6^\circ\).
• Second Quadrant: Subtract from \(180^\circ\), e.g., \(180^\circ - 63.4^\circ \approx 116.6^\circ\).

These values are then rounded as needed, here to the nearest \(10^\circ\), resulting in \(120^\circ\) and \(300^\circ\). Understanding reference angles is vital for determining correct and accurate angle solutions across different quadrants where trigonometric functions take negative or positive values.