Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Give solutions over the interval \([0,2 \pi)\) as approximations to the nearest hundredth when exact values cannot be determined. You may need to use the quadratic formula. Give approximate answers in Exercises \(59-64\) to the nearest tenth of a degree over the interval \(\left[0^{\circ}, 360^{\circ}\right)\) $$\cos ^{2} x-2 \cos x+3=0$$

Step-by-Step Solution

Answer

No real solutions exist for the equation over the interval \([0, 2\pi)\).

Step 1: Identify the Form of the Quadratic

Recognize that the equation \( \cos^2 x - 2\cos x + 3 = 0 \) is a quadratic in terms of \( \cos x \). We will solve for \( \cos x \) as the variable in this equation.

Step 2: Apply the Quadratic Formula

The quadratic formula is \( ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -2 \), and \( c = 3 \). Substitute these into the formula: \[ \cos x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(3)}}{2(1)} \].

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \), which is \((-2)^2 - 4(1)(3) = 4 - 12 = -8 \). Since the discriminant is negative, there are no real roots for \( \cos x \).

Step 4: Conclude That No Real Solutions Exist

Since the quadratic equation \( \cos^2 x - 2\cos x + 3 = 0 \) has no real roots due to the negative discriminant, there are no real solutions for \( x \) in the interval \([0, 2\pi)\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful type of math used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \).This formula helps find the possible values of \( x \) by using the equation:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's a quick breakdown of what each part means:

• \( a \), \( b \), and \( c \) are coefficients of the quadratic equation.
• \( \pm \) signifies that there may be two solutions: one by adding, and one by subtracting.

It can solve any quadratic equation, as long as you know these coefficients. Remember to replace \( x \) with the actual variable you are working with in the problem, like \( \cos x \) in this exercise.

Real Solutions

In the context of equations, real solutions refer to the values that satisfy the equation and are found on the real number line. With quadratic equations, real solutions depend heavily on the value of the discriminant (more on that below).

• If the discriminant is positive, it results in two distinct real solutions.
• If it is zero, there is exactly one real solution (or two identical real solutions).
• If it is negative (as in our exercise), there are no real solutions, only complex ones.

Real solutions are important because they represent achievable, actual values either visually on a graph, or in practical scenarios like measurements or money. In our example, we found out that there are no real solutions, which means the equation does not touch or intersect the x-axis.

Discriminant

The discriminant in a quadratic equation, denoted as \( b^2 - 4ac \), provides valuable information about the nature of the solutions.It tells us if the solutions are real and distinct, real and repeated, or complex (non-real).To calculate the discriminant:

• Square the coefficient of the middle term (\( b^2 \)).
• Subtract the product of four times the first coefficient and the constant term (\( 4ac \)).

For the equation \( \cos^2 x - 2\cos x + 3 = 0 \), the discriminant calculation gave us \( -8 \).Since it is negative, it signals that there are no real solutions for the equation, implying the roots are complex numbers.

Cosine Function

The cosine function, often represented as \( \cos x \), is a fundamental trig function that relates to the angles of a right triangle.It is used widely in trigonometry to connect angles and side lengths.Here are a few key points about the cosine function:

• It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
• It is periodic, meaning it repeats its values in regular intervals, specifically every \( 2\pi \).
• It ranges between \(-1\) and \(1\) for any angle.

In trigonometric equations like \( \cos^2 x - 2\cos x + 3 = 0 \), we solve for \( \cos x \) as part of our quadratic approach.Even if there are no real solutions, understanding how cosine functions behave helps in analyzing angles in other mathematical and physical contexts.