Problem 45 - Precalculus Mathematics for Calculus

Question

Solve the given equation. $$2 \cos ^{2} \theta-7 \cos \theta+3=0$$

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Step 1: Recognize the Form

This equation resembles a quadratic equation, but with \( \cos \theta \) as the variable. It can be rewritten as \( 2x^2 - 7x + 3 = 0 \), where \( x = \cos \theta \).

Step 2: Apply the Quadratic Formula

The quadratic equation \( 2x^2 - 7x + 3 = 0 \) can be solved using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 2 \), \( b = -7 \), and \( c = 3 \).

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \[ (-7)^2 - 4 \times 2 \times 3 = 49 - 24 = 25 \]

Step 4: Solve for \( x \)

Using the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{25}}{2 \times 2} = \frac{7 \pm 5}{4} \]This gives us two potential solutions: 1. \( x = \frac{12}{4} = 3 \)2. \( x = \frac{2}{4} = 0.5 \)

Step 5: Check Validity of Solutions

Since \( x \) represents \( \cos \theta \), we know \( -1 \leq \cos \theta \leq 1 \). Therefore, only \( x = 0.5 \) is valid because \( x = 3 \) is outside this range.

Step 6: Solve for \( \theta \)

With \( \cos \theta = 0.5 \), find the values of \( \theta \) in the standard range \( 0 \leq \theta < 2\pi \). The angles are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).

Key Concepts

Quadratic Formula

Understanding how to solve a quadratic equation is crucial when dealing with certain trigonometric equations. The quadratic formula is a reliable tool that helps find the roots of any quadratic equation, which has the general form \( ax^2 + bx + c = 0 \). The formula itself is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The part under the square root, known as the discriminant \( b^2 - 4ac \), is essential as it determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If zero, the equation has exactly one real root.
• If negative, the roots are complex and not real.

In our specific trigonometric equation, we apply the quadratic formula to solve for the variable \( x \), which represents \( \cos \theta \). Each solution for \( x \) then leads us one step closer to finding the values of \( \theta \).

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a fundamental trigonometric function that maps an angle to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. It is particularly important in this problem because \( \cos \theta \) serves as the variable in our quadratic equation.
The cosine function has values ranging from \(-1\) to \(1\), which means any solutions derived from substituting it back into our equation must fall within this range:

• \( \cos \theta = 0.5 \), which is valid.
• \( \cos \theta = 3 \), which is outside the allowable range, thus invalid.

Knowing these bounds helps us eliminate extraneous solutions and focus on those that make sense within the context of the problem.

Angle Solutions

After determining the valid value for \( \cos \theta \), the next step is to solve for \( \theta \) itself. For \( \cos \theta = 0.5 \), we need to find angles \( \theta \) that satisfy this equation within the range of 0 to \( 2\pi \) radians.
To find these specific angles, we consider where the angle on the unit circle corresponds to a cosine value of 0.5. These angles occur at:

• \( \theta = \frac{\pi}{3} \)
• \( \theta = \frac{5\pi}{3} \)

Both of these angles fall within the standard interval for trigonometric solutions. Ensuring that the answers lie within this range is important for problems where you are considering one full rotation of a circle, from 0 to \( 2\pi \), corresponding to 0 to 360 degrees.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the involved variables where both sides of the equation are defined. These identities are useful tools for simplifying trigonometric expressions and solving equations. While this specific problem involves solving a quadratic form, understanding identities can deepen your understanding and assist in more complex problems.
Some commonly used trigonometric identities include:

• Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Angle sum and difference identities, such as \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \).
• Double angle identities, such as \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \).

Understanding these can help when you encounter more complicated trigonometric equations and need to manipulate or simplify expressions to make them more manageable. They serve as the backbone for developing deeper insights into the behavior of trigonometric functions.