Problem 10 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(\theta = \frac{\pi}{4} + 2n\pi, \frac{3\pi}{4} + 2n\pi, n \in \mathbb{Z}\)

Step 1: Use the Double Angle Formula

The given equation is \[\cos 2\theta = \cos^2 \theta - \frac{1}{2}\]. We know that the double angle formula for cosine is \[\cos 2\theta = 2\cos^2\theta - 1\]. To make both sides of our given equation look similar, use this identity on the left-hand side: \[2\cos^2\theta - 1 = \cos^2\theta - \frac{1}{2}\].

Step 2: Rearrange the Equation

Subtract \(\cos^2 \theta\) from both sides: \[2\cos^2\theta - 1 - \cos^2 \theta = -\frac{1}{2}\].This becomes \[\cos^2 \theta - 1 = -\frac{1}{2}\].

Step 3: Simplify the Equation

Add 1 to both sides to isolate \(\cos^2 \theta\):\[\cos^2 \theta = -\frac{1}{2} + 1\].Simplifying the right side gives \[\cos^2 \theta = \frac{1}{2}\].

Step 4: Solve for \(\theta\)

Take the square root of both sides:\[\cos \theta = \pm \frac{\sqrt{2}}{2}\].The values of \(\theta\) that satisfy this condition are:\[\theta = \frac{\pi}{4} + 2n\pi,\ \theta = -\frac{\pi}{4} + 2n\pi \], and using symmetry across the x-axis, the additional solutions are \[\theta = \frac{3\pi}{4} + 2n\pi,\ \theta = -\frac{3\pi}{4} + 2n\pi \], for integers \(n\).

Key Concepts

Double Angle Formula

The double angle formula is an essential tool in trigonometry. It helps us express trigonometric functions at double angles in terms of single angles. The double angle formula for cosine is especially useful and can be written as:

• \(\cos 2\theta = 2\cos^2\theta - 1\)

This formula can simplify complex equations and make them easier to solve. It changes a representation that might be hard to work with into one that has clear terms involving only one angle.
By applying this formula strategically, you can break down complex trigonometric problems into simpler components.
In our example, we used the formula to transform the left side of the equation, allowing it to be analyzed and manipulated more effectively. Understanding this formula is crucial as it is widely used in solving various trigonometric equations.

Solving Trigonometric Problems

Solving trigonometric problems involves using identities and algebraic techniques to find unknown angles or expressions. The process can often be broken down into straightforward steps:

• Identify the trigonometric identities relevant to the problem.
• Use algebraic manipulation to simplify or rearrange the equation.
• Find a way to isolate the desired variable or function.
• Calculate or derive the solution using known trigonometric values or further algebraic manipulation.

In the given exercise, the process started with recognizing which identities and formulas to apply, such as the double angle formula.
This insight allowed a seemingly complex problem to become manageable, as we transformed and simplified the equation step-by-step.
With practice, solving such trigonometric problems can become intuitive and methodical.

Cosine Function

The cosine function, denoted as \(\cos \theta\), is one of the fundamental trigonometric functions. It is defined in the context of a right triangle as the ratio of the adjacent side to the hypotenuse. Cosine is significant in both geometry and calculus owing to its circular and periodic properties.
It is important to know that the cosine function has a range between -1 and 1, and its period is \(2\pi\), meaning that it repeats its values every \(2\pi\).
In our exercise, the value \(\cos \theta = \pm \frac{\sqrt{2}}{2}\) is extracted from our simplified equation.
This highlights the cosine values when the angles \(\theta = \frac{\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4},\) and \(-\frac{3\pi}{4}\) were found, demonstrating the periodic nature of the function. Understanding the cosine function gives insight into the behavior of waves, oscillations, and rotations in various fields.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. These identities involve trigonometric functions and are essential in simplifying and solving trigonometric equations.

• Common identities include the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\).
• The double angle formulas, like \(\cos 2\theta = 2\cos^2\theta - 1\), introduced earlier.
• Sum and difference formulas, product-to-sum formulas, and more.

These identities are often used to transform and reduce trigonometric expressions to more convenient forms. They are indispensable tools in both basic and advanced trigonometry.
In our solution process, using the double angle formula was a key step, as it allowed us to express \(\cos 2 \theta \) in terms of \(\cos^2 \theta \).
This simplification is often the key to turning a complex equation into a solvable one. Knowing these identities and when to use them can make solving trigonometric equations more straightforward and efficient.