Problem 46 - Algebra 2

Question

Simplify each expression. $$ 2 \cos ^{2} \theta-\cos 2 \theta $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(1\).

Step 1: Identify given expression and use double angle identity

First understand the equation given, which is \(2\cos^2(\theta) - \cos(2\theta)\). Now we can make use of the double angle identity of \(\cos(2\theta)\) which has two possible forms. For this particular case, choose \(\cos(2\theta) = 2\cos^2\theta - 1\), to simplify the equation as we have \(\cos^2\theta\) term in the original equation. Replace \(\cos(2\theta)\) with \(2\cos^2\theta - 1\) in the original equation.

Step 2: Substitute the identity into the given expression

Substitute \(\cos(2\theta) = 2\cos^2\theta - 1\) in the given equation. So the equation becomes, \(2\cos^2(\theta) - (2\cos^2\theta - 1)\).

Step 3: Simplify the equation

Now simplify the equation. This results in \(2\cos^2(\theta) - 2\cos^2\theta + 1\), which will simplify to \(1\) upon combining similar terms.

Key Concepts

Double Angle Formulas

The double angle formulas in trigonometry are crucial for simplifying expressions and solving equations that involve trigonometric functions. These formulas express trigonometric functions of double angles, like \(2\theta\), in terms of single angles, such as \(\theta\). In this exercise, we focused on the double angle formula for cosine:

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

This formula is particularly useful because it allows us to express \(\cos(2\theta)\) using the square of \(\cos(\theta)\). By substituting this identity into the original equation, you can simplify complex trigonometric expressions. Not only does this make calculations more straightforward, but it also helps in solving trigonometric equations efficiently.

Trigonometric Simplification

Trigonometric simplification aims to transform complex trigonometric expressions into simpler forms, making them easier to work with or solve. This involves applying trigonometric identities, like the double angle formulas, to reduce terms.
In our exercise, we simplified the expression \(2\cos^2(\theta) - \cos(2\theta)\) by substituting the identity \(\cos(2\theta) = 2\cos^2\theta - 1\).

• First, replace \(\cos(2\theta)\) so that the entire expression is in terms of \(\cos^2(\theta)\).
• Then, simplify the terms which involve subtracting similar expressions, leading to a much simpler result.

The expression simplifies to \(1\) after eliminating the \(2\cos^2(\theta)\) terms, which is a textbook example of how the right trigonometric identity can significantly ease the problem-solving process.

Cosine Function

The cosine function is one of the fundamental trigonometric functions used in various branches of mathematics and physics. It is often represented with the notation \(\cos\), describing the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
The cosine function is periodic, with a period of \(2\pi\), meaning its values repeat every \(2\pi\). An important aspect of cosine in trigonometry, as shown in the exercise, is how its identities, like the double angle formula, can be leveraged to simplify complex expressions. When combined with identities like \(\cos(2\theta)\), these functions become powerful tools for simplifying expressions and solving equations.

• Recognizing these patterns and relationships is key to mastering trigonometry.
• For our solution, the focus was on utilizing square identities effectively, ensuring clean and reduced results in computations.

Understanding these foundational elements of\(\cos\theta\) helps produce clear solutions and deepens comprehension of how trigonometric identities work.