Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 63-68: Approximate, to the nearest 10', the solutions of the equation in the interval \(\left[0^{\circ}, 360^{\circ}\right)\). $$ 2 \tan ^{2} x-3 \tan x-1=0 $$

Step-by-Step Solution

Answer

The solutions, approximated to the nearest 10 degrees, are: \(60^{\circ}, 240^{\circ}, 170^{\circ}, 350^{\circ}\).

Step 1: Identify the Type of Equation

The given equation is a quadratic equation in terms of \( \tan x \). It is of the form \( ax^2 + bx + c = 0 \), where \( a = 2 \), \( b = -3 \), and \( c = -1 \).

Step 2: Determine the Roots Using the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \( \tan x \). Substituting the values, we get the roots:\[t \tan x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2}\]Simplifying inside the square root:\[\tan x = \frac{3 \pm \sqrt{9 + 8}}{4} = \frac{3 \pm \sqrt{17}}{4}\]So the quadratic formula gives us two potential expressions for \( \tan x \):\[ \tan x = \frac{3 + \sqrt{17}}{4} \] and \[ \tan x = \frac{3 - \sqrt{17}}{4} \].

Step 3: Calculate Numeric Values for Roots

Calculate the decimal values for \( \tan x \):First root: \( \tan x_1 = \frac{3 + \sqrt{17}}{4} \approx 1.78 \).Second root: \( \tan x_2 = \frac{3 - \sqrt{17}}{4} \approx -0.28 \).

Step 4: Find All Possible Angles for Each Root

Convert the decimal \( \tan x \) values to angle measures in degrees:For \( \tan x_1 \approx 1.78 \): Angles \( x \approx 60^{\circ} \) (first quadrant) and \( x \approx 240^{\circ} \) (third quadrant), considering the periodicity of tangent (\(180^{\circ}\)).For \( \tan x_2 \approx -0.28 \): Angles \( x \approx 350^{\circ} \) (fourth quadrant) and \( x \approx 170^{\circ} \) (second quadrant).

Step 5: Approximate Solutions to Nearest 10 Degrees

Round each angle to the nearest 10 degrees. For \( \tan x_1 \), the angles \(60^{\circ}\) and \(240^{\circ}\).For \( \tan x_2 \), the angles \(350^{\circ}\) and \(170^{\circ}\).This gives the solutions: \(60^{\circ}, 240^{\circ}, 170^{\circ}, 350^{\circ}\).

Key Concepts

Quadratic Formula

Understanding the quadratic formula is crucial for solving equations of the form \( ax^2 + bx + c = 0 \). In our given equation, \( 2 \tan^2 x - 3 \tan x - 1 = 0 \), we identify \( a = 2 \), \( b = -3 \), and \( c = -1 \). The quadratic formula itself is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

To apply this formula, you substitute the coefficients \( a \), \( b \), and \( c \) into it. This gives two potential solutions for \( x \), as there's a "plus-minus" in the formula.

• The expression under the square root, \( b^2 - 4ac \), is called the discriminant. It tells us about the nature of the roots.
• If the discriminant is positive, like in our problem where it was \( \sqrt{17} \), there will be two distinct real roots.

By calculating the actual values, we can turn these roots into useful answers for trigonometric expressions.

Periodic Functions

Trigonometric functions, like the tangent function, are known as periodic functions. This means they repeat their values in regular intervals. The tangent function, \( \tan(x) \), specifically, has a period of \( 180^\circ \) or \( \pi \) radians.

This periodic nature is key when solving equations as it helps in identifying all possible angles that satisfy the equation within a given interval. For the interval \([0^\circ, 360^\circ)\), the solutions you find will repeat every \( 180^\circ \).

• For a root with \( \tan(x) = 1.78 \), if one solution is \( 60^\circ \), another valid solution is \( 60^\circ + 180^\circ = 240^\circ \).
• Similar logic applies to other roots and solutions, such as \( -0.28 \), yielding \( 350^\circ \) and \( 170^\circ \).

Recognizing periodicity ensures we don't miss any angles that fit the given conditions within the specified range.

Tangent Function

The tangent function, \( \tan(x) \), is one of the basic trigonometric functions. It's defined as the ratio of the sine of an angle to the cosine of the same angle: \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

It's particularly useful in this exercise because we have a quadratic in terms of \( \tan(x) \). Due to its periodicity and symmetry:

• The tangent function has an infinite range but is undefined where \( \cos(x) = 0 \).
• This occurs at \( x = 90^\circ + 180^\circ n \), where \( n \) is an integer.

Understanding these properties helps us know where to look for solutions and anticipate any vertical asymptotes (undefined points) which can affect solution sets. This function's behavior is essential for determining all possible angles that work with an equation like \( 2 \tan^2 x - 3 \tan x - 1 = 0 \).

Angle Approximation

Approximating angles is essential when our results are not whole numbers. In our equation, we derived values \( \tan x \approx 1.78 \) and \( \tan x \approx -0.28 \). To find the angles, we utilize inverse tangent functions: \( \arctan \), which gives us specific angles in degrees.

• For a \( \tan(x) \) value of \( 1.78 \), \( \arctan(1.78) \approx 60.41^\circ \).
• For \( -0.28 \), \( \arctan(-0.28) \approx -15.64^\circ \), adjusted to the appropriate quadrant as \( 350^\circ \).

Once angles are calculated, approximate these to the nearest \( 10^\circ \), making the final answers user-friendly for practical application. Always remember that when dealing with trigonometric equations, checking your calculator settings for degree mode versus radians is crucial to avoid errors. Developing the skill of angle approximation aids in successfully interpreting and implementing solutions from trigonometric equations into real-world scenarios.