Problem 154 - Precalculus Essentials

Question

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. I solved \(\cos \left(x-\frac{\pi}{3}\right)=-1\) by first applying the formula for the cosine of the difference of two angles.

Step-by-Step Solution

Answer

The statement does not make sense because there's no need to apply the cosine difference formula to solve the given equation. You just need to find the values for which the cosine function equals -1.

Step 1: Examine the statement

The statement is: 'I solved \( \cos \left(x-\frac{\pi}{3}\right)=-1 \) by first applying the formula for the cosine of the difference of two angles.' This refers to the cosine difference formula \( \cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b) \).

Step 2: Analyze the use of the cosine difference formula

The use of the cosine difference formula might not be appropriate in this case. The formula is useful when we want to simplify expressions or prove identities, but here we already have a simple equation to solve. We just need to find for what x-value the cosine function equals -1.

Step 3: Check values where cosine equals -1

We know that the cosine function equals -1 at \( x=\pi \) plus any multiple of \( 2\pi \), therefore the solution to the equation \( \cos \left(x-\frac{\pi}{3}\right)=-1 \) should be \( x=\pi + \frac{\pi}{3} \) plus any multiple of \( 2\pi \). We can confirm this by substituting these values into the given equation and check if it holds true.

Key Concepts

Trigonometric Equations

Trigonometric equations involve unknowns that are angles and are solved using trigonometric functions like sine, cosine, and tangent. One primary goal in solving such equations is finding all possible angles that satisfy the equation, often within a specified interval.
When dealing with cosine, a key characteristic to remember is its periodic nature, meaning the values of cosine repeat at regular intervals, known as its period. The period of the cosine function is \(2\pi\).
Here are a few steps to keep in mind when solving trigonometric equations involving cosine:

• Identify the fundamental period of the cosine function, which is \(2\pi\).
• Find the principal value, the smallest non-negative solution to the equation, if it's within the defined interval.
• Check for additional solutions by considering the periodic nature of the function, typically by adding or subtracting full periods of \(2\pi\).

Solving these equations requires recognizing the special angles where cosine takes extreme values, like 0, -1, and 1, and understanding how they repeat alongside the period of the function.

Angle Subtraction Identity

The angle subtraction identity, specifically for cosine, is a powerful tool in trigonometry. It is used to expand the cosine of the difference between two angles and is represented by the formula:
\[\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)\]
This identity allows us to break down complex expressions into simpler forms.
In the context of solving equations, like \(\cos(x - \frac{\pi}{3}) = -1\), it might be tempting to use this identity. However, its true application shines in proofs or transforming expressions for further simplification, rather than in directly solving equations.

• It transforms expressions involving the difference of two angles into a sum, making them easier to manage.
• Helps in deriving other trigonometric identities and simplifying expressions in terms of known values.

It’s important to note that unless an identity directly simplifies the equation or helps factor it further, applying it might not always yield the quickest path to a solution when solving trigonometric equations.

Cosine Function Properties

The cosine function, vital in trigonometry, has distinct characteristics that help in solving equations effectively. Understanding these properties assists in recognizing patterns and deriving solutions quickly.
Key properties of the cosine function include:

• Range: Its values range from -1 to 1.
• Periodicity: Repeats every \(2\pi\) radians, making its periodicity a key factor to consider in finding multiple solutions.
• Extreme Values: Cosine reaches -1 at odd multiples of \(\pi\) and 1 at even multiples of \(\pi\).

These properties imply that if an equation involves cosine equaling -1, it occurs at all angles of the form \(x = \pi + 2k\pi\) where \(k\) is an integer.
Such knowledge enables us to swiftly determine that if \(\cos(y) = -1\), then \(y\) must be odd multiples of \(\pi\), significantly narrowing down the search for solutions in any trigonometric equation. Recognizing these fundamental properties aids in effectively navigating equations and utilizing solutions in real-world applications.