Problem 72 - Precalculus with Limits

Question

In Exercises 71 - 74, simplify the expression algebraically and use a graphing utility to confirm your answer graphically. \( \cos(\pi + x) \)

Step-by-Step Solution

Answer

The simplified form of \( \cos(\pi + x) \) is \( -\cos(x) \)

Step 1: Use the cosine of sum formula

We commence off by using the sum of angles formula for cosine. which is \( \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \). Here \( a=\pi, b=x \). So, \( \cos(\pi + x) = \cos(\pi) \cos(x) - \sin(\pi) \sin(x) \).

Step 2: Substitute the values

The values for cosine and sine at \( \pi \) is well known. \( \cos(\pi) = -1 \), and \( \sin(\pi) = 0 \). Substitute these values into the formula. Therefore, \( \cos(\pi + x) = -1 \cos(x) - 0 \).

Step 3: Simplify the expression

After substitution, the formula can be further simplified as \( \cos(\pi + x) = - \cos(x) \).

Step 4: Confirm the Result Graphically

To confirm the result graphically, plot \( y = \cos(\pi + x) \) and \( y = -\cos(x) \) in the same set of axis. The two graphs should coincide indicating that \( \cos(\pi + x) = - \cos(x) \).

Key Concepts

Cosine of Sum Formula

The cosine of sum formula is a valuable trigonometric identity that simplifies the process of finding the cosine of the sum of two angles. It is given as: \[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]This formula allows you to break down the cosine of a combined angle into more manageable parts, using basic trigonometric functions of the individual angles, \(a\) and \(b\).
In our problem, we apply this formula to \( \cos(\pi + x) \). Here, \(a = \pi\) and \(b = x\). When we plug these values into the formula, we have: \[ \cos(\pi + x) = \cos(\pi) \cos(x) - \sin(\pi) \sin(x) \]Knowing common angle values, \( \cos(\pi) = -1 \) and \( \sin(\pi) = 0 \). With these, the expression becomes:\[ \cos(\pi + x) = -1 \cdot \cos(x) - 0 \cdot \sin(x) \] This simplifies to \( \cos(\pi + x) = -\cos(x) \).
This identity simplifies equations involving transformations and rotations, making them easier to solve or visualize.

Graphical Verification

Graphical verification is a useful approach in mathematics, allowing us to visually confirm algebraic results. By graphing the expressions involved in our trigonometric identity, we can check whether they behave as expected across different values.
For the expression \( \cos(\pi + x) \), our simplification shows that \( \cos(\pi + x) = -\cos(x) \). To verify this graphically, plot both \( y = \cos(\pi + x) \) and \( y = -\cos(x) \) using a graphing tool.

• Observe how the graphs overlay perfectly. This indicates they are indeed equivalent.
• Ensure a range of \(x\) values to see a well-defined periodic behavior.

This visual overlap supports the algebraic simplification, reinforcing the validity of our solution. Graphical verification is an excellent means of cross-checking results, especially for complex identities.

Angle Addition Identities

Angle addition identities, such as those for sine and cosine, enable us to simplify trigonometric expressions and solve equations involving compound angles. These identities encompass sine and cosine for sum and difference of angles.
Angle addition identity for cosine is:\[ \cos(a + b) = \cos(a) \cos(b) - \sin(a) \sin(b) \]Similarly, there is a corresponding form for sine:\[ \sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b) \]Using these identities helps to transform angles into simpler components, aiding in solving trigonometric equations.
Knowing these identities is essential for interpreting and simplifying the behavior of angles in various physical and theoretical contexts. Whether calculating rotations, oscillations, or resonating frequencies, these identities provide the tools needed for simplification and solution.