Problem 6 - Calculus Single Variable

Question

Calculate the given expression without using a calculator. \(\cos (0)-\cos (\pi)\)

Step-by-Step Solution

Answer

The value of the expression is 2.

Step 1: Recall Key Trigonometric Values

Remember that cosine is an even function, meaning \( \cos(-x) = \cos(x) \). The cosine of 0 radians is \( \cos(0) = 1 \) and the cosine of \( \pi \) radians is \( \cos(\pi) = -1 \).

Step 2: Substitute Values into the Expression

Substitute the known values of \( \cos(0) \) and \( \cos(\pi) \) into the expression \( \cos (0)-\cos (\pi) \). We have: \( \cos(0) = 1 \) and \( \cos(\pi) = -1 \).

Step 3: Simplify the Expression

Substitute the values: \( \cos (0)-\cos (\pi) = 1 - (-1) \). Simplify this to get: \( 1 + 1 = 2 \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \( \cos(x) \). It relates the angle \( x \) to the ratio of the adjacent side over the hypotenuse in a right-angled triangle. Cosine is integral to understanding wave patterns and circular motion.

Some important properties of the cosine function include:

• Periodic Nature: Cosine is periodic with a period of \( 2\pi \), meaning \( \cos(x) = \cos(x + 2\pi) \).
• Domain and Range: The domain of cosine is all real numbers, and its range is \([-1, 1]\).
• Symmetrical Graph: The graph of the cosine function is a smooth, oscillating curve that repeats every \( 2\pi \).

This function plays a big role in solving problems related to angles and distances, such as in the given exercise.

Even Function

An even function has symmetry about the y-axis, meaning that its graph is mirrored over the vertical axis. Mathematically, a function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \) in its domain. This property simplifies calculations by letting us know that the function's value is the same for opposite arguments.

Cosine is a classic example of an even function because it satisfies this condition:

• \( \cos(-x) = \cos(x) \)

In practical terms, for any given angle, flipping the angle to its negative does not alter the cosine value. This feature is useful in symmetry checks and simplifying expressions, like the exercise solution.

Key Trigonometric Values

Key trigonometric values for angles like \( 0 \, \text{and} \, \pi \) are essential for solving trig problems quickly and accurately. These values are often memorized because they appear frequently:

• \( \cos(0) = 1 \)
• \( \cos(\pi) = -1 \)
• \( \sin(0) = 0 \)
• \( \sin(\pi) = 0 \)

These basic values arise from the unit circle, where \( 0 \) radians is at the rightmost point and \( \pi \) radians is directly opposite on the left. Recognizing these standard values lets us bypass more complex computations, ensuring efficient and correct resolutions.

Simplifying Expressions

Simplifying expressions involves breaking down complex trigonometric expressions into simpler parts in order to find solutions more easily. In the exercise, simplifying the expression \( \cos(0) - \cos(\pi) \) means substituting the known cosine values and performing arithmetic:

• Substitute \( \cos(0) = 1 \) and \( \cos(\pi) = -1 \).
• Expression becomes \( 1 - (-1) \).
• Simplify to \( 1 + 1 = 2 \).

This process reduces potential errors and enhances understanding through clear, logical steps, allowing problems to be solved swiftly and correctly.