Problem 36 - Precalculus with Calculus Previews

Question

Justify the given statement with one of the properties of the trigonometric functions. $$ \cos 16.8 \pi=\cos 14.8 \pi $$

Step-by-Step Solution

Answer

The statement is justified by the periodicity property of cosine (\(\cos(\theta) = \cos(\theta + 2n\pi)\)).

Step 1: Identify the Trigonometric Property

We need to justify why \(\cos 16.8\pi = \cos 14.8\pi\). A key trigonometric property that can be applied here is the periodicity of cosine. The cosine function is periodic with a period of \(2\pi\). This means that \(\cos(\theta) = \cos(\theta + 2k\pi)\) for any integer \(k\).

Step 2: Identify the Difference in Angles

To use the periodicity, we need to ensure that the difference between \(16.8\pi\) and \(14.8\pi\) is a multiple of \(2\pi\). Calculate the difference: \(16.8\pi - 14.8\pi = 2\pi\), which confirms that the difference is indeed a multiple of \(2\pi\).

Step 3: Apply the Cosine Periodicity Property

Based on the periodicity property of cosine, since the difference between \(16.8\pi\) and \(14.8\pi\) is \(2\pi\), we have \(\cos(16.8\pi) = \cos(14.8\pi)\) because the cosine function returns to the same value every \(2\pi\).

Step 4: Conclude with the Justification

The given statement \(\cos 16.8\pi = \cos 14.8\pi\) is justified using the periodicity property of the cosine function, which states that for any angle \(\theta\), \(\cos(\theta) = \cos(\theta + 2n\pi)\) where \(n\) is an integer.

Key Concepts

Cosine Periodicity

The concept of cosine periodicity is fundamental in understanding how trigonometric functions behave over intervals. A periodic function is one that repeats its values in regular intervals or periods. For the cosine function, this period is exactly

• \(2\pi\): meaning every time you add \(2\pi\) to the angle, the cosine function produces the same result.

To say this more plainly, the cosine of any angle \(\theta\) is the same as the cosine of \(\theta + 2k\pi\), where \(k\) is any integer.
For example, if you know \(\cos(\theta)\), you instantly know \(\cos(\theta + 2\pi)\), \(\cos(\theta + 4\pi)\), and so on.This repetitive property is particularly useful in trigonometric identities and equations. It simplifies many calculations and is key to solving problems where angles recur or accumulate over cycles, like in rotations or waves.

Cosine Function

The cosine function is one of the primary trigonometric functions, alongside sine and tangent. It is defined using a right triangle or as a function on the unit circle. In the context of a right triangle, the cosine of an angle \(\theta\) is the ratio of the length of the adjacent side to the hypotenuse. Mathematically, this is expressed as

• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)

When moving to the unit circle, we consider a circle with a radius of 1. The cosine function becomes the \(x\)-coordinate of the point where the terminal side of the angle \(\theta\) intersects the circle. As the angle \(\theta\) changes, the value of \(\cos(\theta)\) represents its horizontal projection.
This beautiful structure and definition make the cosine function prevalent in various fields, such as engineering and physics, where analyzing waves and oscillations is essential.

Angle Difference Calculation

An important aspect of many trigonometric problems is calculating angle differences. This involves finding the result of subtracting one angle from another, which is crucial for identifying properties related to periodic functions. In the given solution, the difference between the angles \(16.8\pi\) and \(14.8\pi\) is calculated as follows:

• \(16.8\pi - 14.8\pi = 2\pi\)

Recognizing this difference as a multiple of the cosine function's period allows us to conclude that the values of the cosine function at these angles are identical.Understanding angle difference calculation is essential for applying periodic properties and solving complex trigonometric problems. It lets us reduce the complexity of trigonometric expressions by focusing on periodicity and symmetry, making analysis more manageable and intuitive.