Problem 46 - Precalculus Essentials

Question

Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$\cos ^{2} \theta-1=0$$

Step-by-Step Solution

Answer

The solutions to the equation within the interval \([0, 2\pi]\) are \(0, \pi\), and \(2\pi\).

Step 1: Manipulate to isolate cosine term

The aim in this step is to isolate the cosine term on one side of the equation. We can start by rewriting the equation as \(\cos^{2}\theta = 1\). By taking the square root on both sides of the equation, remember, we must consider both the positive and negative roots which looks like: \(\cos\theta = \pm 1\).

Step 2: Find the values of theta solving for cosine with \(\pm 1\)

Next, find the corresponding values of \(\theta\) when \(\cos\theta = 1\) and \(\cos\theta = -1\) within the interval \([0, 2\pi]\). In this case, cosine of \(\theta\) is 1 at \(\theta = 0\) and \(\theta = 2\pi\), and cosine of \(\theta\) is -1 at \(\theta = \pi\).

Step 3: List all the solutions

The final step is to list all the solutions within the given interval. The solutions should include \(0, \pi\), and \(2\pi\).

Key Concepts

Quadratic Forms

Quadratic forms often appear in trigonometric equations, making them crucial for solving problems involving periodic functions like sine or cosine. In mathematical terms, a quadratic form resembles the equation of general form \(ax^2 + bx + c = 0\). In the context of trigonometric equations, this may manifest similarly, but with trigonometric functions instead of simple variables. For example, the equation \(\cos^2\theta - 1 = 0\) is analogous to a quadratic equation, where \(\cos^2\theta\) plays the role of \(x^2\).

• This structure helps in recognizing and approaching the equation using known strategies for quadratics.
• Rewriting equation in standard form helps to use methods like factoring or using quadratic formula.

In the example given, isolating \(\cos^2\theta\) on one side leads us to solve like traditional quadratic equations, where understanding the roots offers solutions for corresponding angles \(\theta\). In this setup, the equation simplifies to distinct values for \(\cos\theta\) indicating fixed points on the unit circle.

Cosine Function

The cosine function is one of the primary trigonometric functions, crucial to resolving problems involving angles and lengths. When solving equations like \(\cos^2\theta = 1\), understanding what each value implies is critical.

• \(\cos\theta\) ranges between -1 and 1, rocking back and forth as the angle revolves around the unit circle.
• At \(\cos\theta = 1\), the angle \(\theta\) is at a position where the terminal point is on the positive x-axis, typically zero or multiples of \(2\pi\).
• On the other hand, \(\cos\theta = -1\) results in angles such as \(\pi\), where the point lies on the negative x-axis.

These positions relate directly to the cosine function's periodic and oscillatory nature and aid in identifying angles corresponding to specific trigonometric values. Moreover, it's pivotal to remember the cosine function's periodicity and symmetry which assist in recognizing similar functions in broader intervals.

Interval Notation

Interval notation is a simple way to represent a range of numbers, which is especially useful in trigonometry when specifying domains like \([0, 2\pi)\). Understanding and correctly using interval notation helps in navigating through solutions over specific ranges.

• An interval \([a, b)\) includes all numbers from \(a\) to \(b\), including \(a\), but excluding \(b\).
• This notation provides a clear boundary to tell what part of the circle or angle range we're considering.

In trigonometric problems, especially when functions are periodic, recognizing \([0, 2\pi)\) implies analyzing angles from \(0\) radians up to, but not including, \(2\pi\). This is significant since angles repeat after \(2\pi\), thus enforcing a specific boundary to avoid redundancy. Proper use of interval notation ensures every solution within the span is identified without repetition.