Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right) .\) Give solutions to the nearest tenth as appropriate. $$4 \cos ^{2} \theta+4 \cos \theta=1$$

Step-by-Step Solution

Answer

The solutions are \( \theta \approx 74.1^\circ \) and \( \theta \approx 285.9^\circ \).

Step 1: Write the Equation in Standard Form

First, start by rewriting the given equation into a standard quadratic form. The given equation is \( 4 \cos^2 \theta + 4 \cos \theta = 1 \). Subtract \( 1 \) from both sides to get: \( 4 \cos^2 \theta + 4 \cos \theta - 1 = 0 \).

Step 2: Let Substitution Simplify the Equation

Introduce a substitution to simplify the expression. Let \( x = \cos \theta \). Then the equation transforms to \( 4x^2 + 4x - 1 = 0 \).

Step 3: Apply the Quadratic Formula

Since the equation \( 4x^2 + 4x - 1 = 0 \) is quadratic, use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = 4 \), and \( c = -1 \).

Step 4: Calculate the Discriminant

First, calculate the discriminant \( b^2 - 4ac \). Here, it is \( 4^2 - 4 \cdot 4 \cdot (-1) = 16 + 16 = 32 \).

Step 5: Solve for x-Values

Use the quadratic formula to solve for \( x \): \[ x = \frac{-4 \pm \sqrt{32}}{8} \] \[ x = \frac{-4 \pm 4\sqrt{2}}{8} \] Which simplifies to \( x = \frac{-1 \pm \sqrt{2}}{2} \).

Step 6: Convert x-Values Back to Theta

Since \( x = \cos \theta \), solve for \( \theta \). We have two solutions: 1. \( \cos \theta = \frac{-1 + \sqrt{2}}{2} \) 2. \( \cos \theta = \frac{-1 - \sqrt{2}}{2} \). Since \( \frac{-1 - \sqrt{2}}{2} \) is less than \( -1 \), it is impossible for the cosine function to take this value. Hence, we proceed only with the first solution.

Step 7: Find the Angle Theta

Convert \( \cos \theta = \frac{-1 + \sqrt{2}}{2} \) to an angle within the interval \([0^\circ, 360^\circ)\). Calculate: \[ \theta = \cos^{-1}\left(\frac{-1 + \sqrt{2}}{2}\right) \approx 74.06^\circ \] and \( \theta = 360^\circ - 74.06^\circ \approx 285.94^\circ \). These are approximately \( \theta = 74.1^\circ \) and \( \theta = 285.9^\circ \).

Step 8: Verify the Solutions

Substitute back to verify solutions. Both \( \theta = 74.1^\circ \) and \( \theta = 285.9^\circ \) satisfy the original equation \( 4 \cos^2 \theta + 4 \cos \theta = 1 \). Hence they are valid.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions used in mathematics. It relates the ratio of the adjacent side to the hypotenuse in a right-angled triangle. When solving trigonometric equations like \( 4 \cos^2 \theta + 4 \cos \theta = 1 \), understanding the properties of the cosine function is essential.

• The cosine of an angle varies between \(-1\) and \(1\).
• It is periodic with a period of \(360^\circ\) or \(2\pi\) radians.
• Placing cosine in its simplest form helps in equation solving, turning into a familiar algebraic format.

In this context, by solving for \(\cos \theta\), we're able to find angles \(\theta\) that align with the solution using the inverse cosine function, resulting in values within the given interval \([0^\circ, 360^\circ)\). Understanding these characteristics aids in predicting and computing possible angle solutions effectively.

Quadratic Formula

The quadratic formula is a powerful tool in algebra used to find the solutions (roots) of quadratic equations of the form \( ax^2 + bx + c = 0 \). Whenever you encounter a trigonometric equation like our cosine example, it can be transformed into a quadratic equation. This transformation lets us utilize the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant

After substitution and simplifications, the equation \( 4\cos^2 \theta + 4\cos \theta - 1 = 0 \) becomes \( 4x^2 + 4x - 1 = 0 \). Here, the coefficients are \(a = 4, b = 4, c = -1\). The quadratic formula allows for the uncovering of possible values of \( x \) (in this case, \( \cos \theta \)), making it an indispensable method in solving quadratic trigonometric equations.

Discriminant

The discriminant of a quadratic equation, found under the square root in the quadratic formula, is crucial in determining the nature and number of the roots. It is given by \( b^2 - 4ac \). For our equation, the discriminant is computed as
\[ 4^2 - 4 \times 4 \times (-1) = 16 + 16 = 32 \]
A positive discriminant, like \(32\), indicates that our equation has two distinct real roots.

• If the discriminant were zero, there would be exactly one real root.
• A negative discriminant would mean the roots are complex (non-real).

This piece of information is vital in understanding that there are indeed valid solutions \(x\)—or \(\cos \theta\) in trigonometric equations—that can be calculated, which will help us in determining the corresponding angle \(\theta\) when mapped back to the trigonometric function.

Principal Values

In trigonometry, solving equations often leads to finding principal values. A principal value is the smallest non-negative angle that satisfies the equation. For the cosine function, when we find \( \cos^{-1} \left( \frac{-1 + \sqrt{2}}{2} \right) \), we acknowledge that this inverse operation will initially give us the principal value.
The process of finding principal values includes:

• Calculating the angle \(\theta\) in the specific range it's defined, usually within \([0^\circ, 180^\circ)\) for \(\cos\).
• Considering additional co-terminal angles, by utilizing periodicity, such as \(360^\circ - \theta\), to cover the complete set that satisfies the original condition.

Working with principal values helps us confidently identify the correct angles within \\([0^\circ, 360^\circ)\) that correspond to our solutions in trigonometric problems. In this case, the calculation resulted in angles approximately: \( \theta = 74.1^\circ \) and \( \theta = 285.9^\circ \). These angles are checked against the initial trigonometric condition through substitution to ensure correctness.