Problem 92 - Precalculus Essentials

Question

Use a calculator to solve each equation, correct to four decimal places, on the interval \([0,2 \pi)\) $$3 \cos ^{2} x-8 \cos x-3=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(3 \cos ^{2} x-8 \cos x-3=0\) in the interval \([0,2\pi)\) are \(x=\frac{2\pi}{3}\) and \(x=\frac{4\pi}{3}\)

Step 1: Identify the Type of the Equation

The given equation resembles a quadratic equation \(ax^2+bx+c=0\) in \(\cos x\) where \(a=3, b=-8\), and \(c=-3\). So, we will solve for \(\cos x\) first.

Step 2: Solve the Quadratic Equation

We can solve the quadratic equation using the quadratic formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). In this context, it becomes \(\cos x=\frac{8\pm\sqrt{(-8)^2-4*3*(-3)}}{2*3}\). After calculation this gives \(\cos x = -0.5, 2\).

Step 3: Solve for x

Since \(\cos x = 2\) is not possible (as -1 < = cos x <=1), we disregard it. Now we have to figure out the values of \(x\) for which \(\cos x = -0.5\). These values within \([0,2\pi]\) are \(x=\frac{2\pi}{3}\) and \(x =\frac{4\pi}{3}\).

Step 4: Verify the Answers

Finally, substitute the obtained values of \(x\) into the given equation to confirm if the it holds true. If the left-hand side equals to the right-hand side of the equation then the obtained values of \(x\) are correct.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and have the general form \(ax^2 + bx + c = 0\). These equations are used commonly to find solutions known as roots. In our exercise, the equation \(3 \cos^2 x - 8 \cos x - 3 = 0\) is structured like a quadratic equation based on the variable \(\cos x\). The key components are:

• \(a = 3\)
• \(b = -8\)
• \(c = -3\)

These coefficients guide the process of finding solutions using different methods like factoring or formulas.
Quadratic equations may have two, one, or no real solutions depending on the discriminant \(b^2 - 4ac\). Each potential solution gives insight into the original problem, and helps to solve the equation using tools like calculators or symbolic manipulation.

Cosine Function

The cosine function is one of the primary trigonometric functions, typically used in contexts involving triangles or periodic phenomena. It is defined on the unit circle with respect to the x-coordinate. In our scenario, the expression \(\cos x\) is essential because it transforms the given trigonometric equation into a quadratic form with \(\cos x\) as the variable.

• The range of the cosine function is between -1 and 1.
• As a periodic function, its uses are extensive in waves and oscillations.
• To solve equations like \(\cos x = -0.5\), reference angles and the unit circle play an important role.

Understanding the periodic and oscillatory nature of the cosine function is crucial when interpreting solutions, especially within specific intervals like \([0,2\pi)\). It signifies calculating angles that satisfy the equation.

Interval Notation

Interval notation is a method of describing a set of numbers along a continuous range. It's a precise way to define the domain or solutions of equations, especially in trigonometry where angles often recur. When dealing with trigonometric problems like ours, interval notation helps simplify solutions.

• The notation \([0, 2\pi)\) means all numbers from 0 to \(2\pi\), including 0 but not \(2\pi\).
• This is used often in contexts where the period of trigonometric functions is relevant.
• It is essential for expressing where solutions actually lie on the trigonometric circle.

Interval notation is beneficial because it explicitly specifies the scope, which eliminates any ambiguity, making it easier to interpret and solve trigonometric equations.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It's given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To apply this formula, the coefficients \(a\), \(b\), and \(c\) must be identified correctly.
In our context, for the equation involving \(\cos x\), plugging in \(a = 3\), \(b = -8\), and \(c = -3\) into the formula yields \(\cos x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot (-3)}}{2 \cdot 3}\).

• This formula helps find real or complex solutions by evaluating the discriminant \(b^2 - 4ac\).
• The negative sign under the square root indicates complex solutions, while a positive or zero provides real ones.
• Understanding and using the quadratic formula is foundational in solving varied mathematical problems efficiently.

Using the quadratic formula ensures a systematic approach to problem-solving where factoring or graphing may not be feasible.