Problem 41

Question

Solve the given equation. $$4 \cos ^{2} \theta-4 \cos \theta+1=0$$

Step-by-Step Solution

Verified

Answer

Solutions are $ \theta = \frac{\pi}{3} + 2n\pi $ or $ \theta = \frac{5\pi}{3} + 2n\pi $ where $ n $ is an integer.

1Step 1: Understand the Equation

The given equation is $4 \cos^{2} \theta - 4 \cos \theta + 1 = 0$. This is a quadratic equation in terms of $ \cos \theta $. To solve it, we treat $ \cos \theta $ as a variable, say $ x $, and rewrite the equation as $ 4x^2 - 4x + 1 = 0 $.

2Step 2: Apply the Quadratic Formula

The general form of a quadratic equation is $ ax^2 + bx + c = 0 $. Here, $ a = 4 $, $ b = -4 $, and $ c = 1 $. The quadratic formula is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. Substitute the values to get $ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{8} $.

3Step 3: Simplify the Discriminant

Simplify $ \sqrt{(-4)^2 - 4 \times 4 \times 1} $ which is $ \sqrt{16 - 16} $, resulting in $ \sqrt{0} $. Thus the expression becomes $ x = \frac{4 \pm 0}{8} $.

4Step 4: Solve for x

Since the discriminant is zero, there is only one solution for $ x $: $ x = \frac{4}{8} = \frac{1}{2} $. Therefore, $ \cos \theta = \frac{1}{2} $.

5Step 5: Find the Angles for $ \cos \theta = \frac{1}{2} $

The cosine function equals $ \frac{1}{2} $ at angles $ \theta = 60^\circ $ or $ \theta = 300^\circ $ (in degrees), or $ \theta = \frac{\pi}{3} $ and $ \theta = \frac{5\pi}{3} $ (in radians).

6Step 6: Complete the Solution

The general solution for $ \cos \theta = \frac{1}{2} $ can also be expressed as $ \theta = 2n\pi \pm \frac{\pi}{3} $ where $ n $ is any integer.

Key Concepts

Quadratic EquationsCosine FunctionRadians and Degrees

Quadratic Equations

Quadratic equations are a fundamental concept in algebra, and they generally take the form $ ax^2 + bx + c = 0 $. These equations involve terms where the variable is raised to the highest power of two. Solving these equations usually demands finding the values of $ x $ that satisfy the equation. To achieve this, we often use the quadratic formula:

$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

Let's break it down:

The formula allows us to calculate the roots of any quadratic equation directly.
The term $ b^2 - 4ac $ is known as the 'discriminant'.
It determines the nature of the roots:

If it is positive, there are two distinct real roots.
If zero, there is exactly one real root.
If negative, the equation has complex roots.

In our cosine function equation, the value of the discriminant was zero, providing exactly one real root. This simplified solving the equation to finding $ x = \frac{1}{2} $.

Cosine Function

The cosine function is one of the primary trigonometric functions. Cosine is the ratio of the adjacent side of a right triangle to its hypotenuse, and it's denoted by $ \cos \theta $. Here are some critical points to understand:

The cosine function is periodic, with a cycle that repeats every $ 2\pi $ radians or 360 degrees.
Values of the cosine function oscillate between -1 and 1.

Graphical Understanding

The graph of the cosine function forms a wave-like pattern that helps visually identify points where the cosine equals specific values.

At $ \theta = 0 $ or any multiples of $ 2\pi $, $ \cos \theta = 1 $.
When $ \theta = \pi/2 $ or its odd multiples, $ \cos \theta = 0 $.
Sustained at -1 at $ \theta = \pi $ or its even multiples.

Knowing these values and the behavior of cosine aids in solving equations like $ \cos \theta = \frac{1}{2} $, pinpointing angles such as 60° or $ \frac{\pi}{3} $.

Radians and Degrees

Radians and degrees are two units for measuring angles. Understanding both is key in trigonometry for converting and solving problems:

Degrees: The full circle is 360 degrees. It's the more common unit used in everyday contexts and initial learning.
Radians: This unit is more prevalent in higher mathematics. A full circle is $ 2\pi $ radians.