Problem 45 - Algebra and Trigonometry

Question

\(39-56 \approx\) Solve the given equation. $$ 2 \cos ^{2} \theta-7 \cos \theta+3=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \( \cos \theta = \frac{1}{2} \), giving angles \( \theta = \frac{\pi}{3} + 2n\pi \) and \( \theta = \frac{5\pi}{3} + 2n\pi \).

Step 1: Identify the type of equation

The equation given is a quadratic equation in terms of \( \cos \theta \). This equation is similar to \( ax^2 + bx + c = 0 \), where \( x = \cos \theta \).

Step 2: Use the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -7 \), and \( c = 3 \). We will substitute these values into the formula.

Step 3: Calculate the discriminant

Calculate \( b^2 - 4ac \). Substituting the values, \((-7)^2 - 4 \times 2 \times 3 = 49 - 24 = 25\).

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

Now, substitute the discriminant back into the quadratic formula: \[ \cos \theta = \frac{-(-7) \pm \sqrt{25}}{2 \times 2} = \frac{7 \pm 5}{4} \]This gives us two solutions for \( \cos \theta \):1. \( \cos \theta = \frac{12}{4} = 3 \)2. \( \cos \theta = \frac{2}{4} = \frac{1}{2} \)

Step 5: Validate possible solutions

Since \( \cos \theta \) can only take values between -1 and 1, \( \cos \theta = 3 \) is not a valid solution. Thus, the only valid solution is \( \cos \theta = \frac{1}{2} \).

Step 6: Find possible angles \( \theta \)

For \( \cos \theta = \frac{1}{2} \), angles \( \theta \) can be \( \theta = \frac{\pi}{3} + 2n\pi \) or \( \theta = \frac{5\pi}{3} + 2n\pi \), where \( n \) is any integer, due to the periodic nature of cosine.

Key Concepts

Quadratic Equation

A quadratic equation is any equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In the context of trigonometric equations, one can treat the cosine function or other trigonometric expressions as variables. This makes it possible to apply methods such as factoring, completing the square, or using the quadratic formula to find solutions.

Here, the equation \( 2 \cos^2 \theta - 7 \cos \theta + 3 = 0 \) is specifically a quadratic in terms of \( \cos \theta \). It is structured just like a typical quadratic equation, reminding us that core algebraic techniques are widely applicable in solving trigonometric equations. Even when dealing with trigonometric functions, recognizing the equation's form can help guide the solution process.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is one of the fundamental trigonometric functions. It represents the x-coordinate of a point on the unit circle as a function of the angle \( \theta \) from the positive x-axis.

• The values of the cosine function range from -1 to 1.
• It is periodic with a period of \( 2\pi \), meaning the function repeats its values every \( 2\pi \) radians.
• It is symmetric about the y-axis, known as an even function.

In the problem, finding \( \cos \theta = \frac{1}{2} \) signifies that the angle \( \theta \) can take on specific values corresponding to where this cosine value occurs on the unit circle. Recognizing and applying periodicity helps list all possible angles given this cosine value.

Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the expression \( b^2 - 4ac \). The discriminant helps determine the nature of the solutions for the quadratic equation.

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution (a repeated root).
• If it is negative, there are no real solutions—instead, they are complex.

In the exercise, the discriminant was calculated as \( 25 \), which is positive. This hints that there are two potential solutions for \( \cos \theta \), though not all may be valid given the valid range of cosine functions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables where they are defined. They are useful for simplifying and solving trigonometric equations.

The periodic and symmetric properties of trigonometric functions are key identities that often help solve equations.
For \( \cos \theta \), important identities include:

• \( \cos(-\theta) = \cos \theta \) (even function identity).
• The cosine of complementary angles: \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \).

Using these, one can derive the values of angles for given cosine values, as done when solving for \( \theta \) in this exercise. Understanding these identities helps anticipate solutions and confirm their validity in the context of infinite periodic angles.