Problem 57 - 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 \(\left[0^{\circ}, 360^{\circ}\right)\) $$2 \cos ^{2} x+2 \cos x=1$$

Step-by-Step Solution

Answer

The approximate solutions are \(x \approx 55.5^{\circ}\) and \(x \approx 304.5^{\circ}\).

Step 1: Simplify the Equation

First, let's rewrite the equation, factoring in common terms. We have:\[ 2\cos^2 x + 2\cos x - 1 = 0. \]We can simplify this by dividing every term by 2 to make calculations easier:\[ \cos^2 x + \cos x - \frac{1}{2} = 0. \]

Step 2: Use the Quadratic Formula

The equation \( \cos^2 x + \cos x - \frac{1}{2} = 0 \) is a quadratic in terms of \( \cos x \). Let \( y = \cos x \). The quadratic formula is\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]Here, \(a = 1\), \(b = 1\), and \(c = -\frac{1}{2}\). Substitute these values into the formula:\[ y = \frac{-1 \pm \sqrt{1^2 - 4(1)(-\frac{1}{2})}}{2(1)}. \]

Step 3: Solve the Discriminant

Calculate the discriminant:\[ 1^2 - 4 \cdot 1 \cdot \left(-\frac{1}{2}\right) = 1 + 2 = 3. \]Thus, the quadratic formula yields two solutions:\[ y = \frac{-1 + \sqrt{3}}{2} \quad \text{and} \quad y = \frac{-1 - \sqrt{3}}{2}. \]

Step 4: Solve for \( \cos x \)

The solutions for \( y = \cos x \) become:\[ \cos x = \frac{-1 + \sqrt{3}}{2} \quad \text{and} \quad \cos x = \frac{-1 - \sqrt{3}}{2}. \]We analyze these to find \( x \). The second solution provides \( \cos x < -1 \), which is impossible, so we discard it.

Step 5: Find Angles in Degree Measure

Using a calculator, solve \( \cos x = \frac{-1 + \sqrt{3}}{2} \) for \( x \):\[ x = \cos^{-1}\left(\frac{-1 + \sqrt{3}}{2}\right). \]Calculate this to find the primary angle within \([0^{\circ}, 360^{\circ})\). The value approximately is:\[ x \approx 55.5^{\circ}. \]Since cosine is positive in the first and fourth quadrants, the possible angles are:\[ x \approx 55.5^{\circ} \quad \text{and} \quad x \approx 360^{\circ} - 55.5^{\circ} = 304.5^{\circ}. \]

Key Concepts

Quadratic Formula

In mathematics, the quadratic formula is a crucial tool for solving quadratic equations—equations of the form \( ax^2 + bx + c = 0 \). By rearranging the equation into the standard form, one can identify the coefficients \( a \), \( b \), and \( c \) and substitute them into the formula:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula gives two solutions, considering the "plus-minus" sign, enabling the determination of the roots of the quadratic equation. Here, the symbol \( \sqrt{} \) points to the square root, and the expression under it is called the discriminant:

• A positive discriminant means two distinct real solutions.
• A zero discriminant results in one real repeated solution.
• A negative discriminant indicates no real solutions.

The quadratic formula simplifies solving complex trigonometric equations that can often be transformed into a quadratic form. As demonstrated in this exercise, it helped solve the trigonometric equation by setting \( y = \cos x \) and applying the formula to find possible values for \( y \) which could then be translated into angles \( x \).

Cosine Function

The cosine function, represented by \( \cos \), is one of the fundamental trigonometric functions. It is crucial for understanding angles and their relationships in a triangle, particularly right triangles.The value \( \cos x \) corresponds to the ratio of the length of the adjacent side to the hypotenuse in a right triangle. It is also important in the unit circle, where it represents the x-coordinate of a given angle's corresponding point.The cosine function has specific properties:

• It is periodic with a period of \(2\pi\), meaning that \( \cos(x + 2\pi) = \cos x \).
• The range is from \(-1\) to \(1\), inclusive.
• It is even, so \( \cos(-x) = \cos x \).

These properties help us deduce that for any angle \( x \), there are usually two angles within a full rotation \([0, 360^{\circ})\) that share the same cosine value. Understanding these properties is why we were able to discard the solution \( \cos x = \frac{-1 - \sqrt{3}}{2} \) in our problem, as it was less than \(-1\), making it invalid.

Angle Approximation

Sometimes exact trigonometric values aren't easy to determine analytically. This is when angle approximation becomes essential. By using a calculator, we approximate angles to get nearly exact values when solving trigonometric equations.In the given exercise, the equation \( \cos^{-1}\left(\frac{-1 + \sqrt{3}}{2}\right) \) was solved to approximate the angle \( x \). Calculators typically use algorithms to compute these approximations to a user-specified accuracy, such as the nearest hundredth.When approximating angles, a few points should be considered:

• Ensure that the angle unit (degrees or radians) is correct as required.
• Be mindful of quadrant considerations, as functions like cosine have symmetry and periodic nature.
• Understand the tolerance of the approximation, acknowledging how close the result is to the exact value.

Through approximation, we were able to establish that the potential angles for \( x \) were approximately \( 55.5^{\circ} \) and its symmetrical counterpart \( 304.5^{\circ} \). This process demonstrates how practical angle approximation is when finding precise values in complex trigonometric solutions.