Problem 40 - Algebra 2

Question

Which statement(s) is (are) true? I. \(\cos \theta=\cos (-\theta)\) \(\quad\) II. \(\cos (\theta+2 \pi)=\cos \theta\) \(\quad\) III. \(\cos \pi=-\cos \pi\) A. I and II only B. II only C. I and III only D. I, II, and III

Step-by-Step Solution

Answer

So, both Statement I and II are true. And Statement III is not true. Hence, the correct option is 'A. I and II only'.

Step 1: Verifying Statement I

The cosine function is an even function meaning it has symmetry about the y-axis. Hence, \(\cos \theta \) equals \(\cos (- \theta)\). Therefore, the first statement is true.

Step 2: Verifying Statement II

The cosine function has a period of \(2\pi\). It means that the cosine value repeats after an interval of \(2\pi\). Hence, \(\cos (\theta + 2\pi) \) equals to \(\cos \theta\). Therefore, the second statement is true.

Step 3: Verifying Statement III

The cosine value of \(\pi\) is -1. Thus, \(\cos \pi\) cannot be equal to \(-\cos \pi \). Thus, the third statement is not true.

Key Concepts

Even Function

An even function is a function that satisfies the condition \( f(x) = f(-x) \) for all values in its domain. This means that the graph of the function is symmetric about the y-axis. The cosine function, \( \cos(x) \), is a perfect example of an even function.
When you evaluate it at a positive angle, say \( \theta \), and its negative counterpart, \( -\theta \), the values will be the same.
This property is significant because it simplifies problems involving angles and reduces calculation errors in trigonometry.

• For example, if you know that \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), then you immediately know that \( \cos(-30^\circ) = \frac{\sqrt{3}}{2} \) without further calculation.
• This symmetry helps in visualizing and predicting behavior in wave functions and signals where cosine might be involved.

Periodicity of Trigonometric Functions

Trigonometric functions are known for their periodic nature. The cosine function, specifically, has a period of \(2\pi\). This means that if you add \(2\pi\) to any angle \(\theta\), the cosine of that angle will remain the same. This is expressed as \( \cos(\theta + 2\pi) = \cos(\theta) \).
The periodicity of cosine allows for the prediction of its value cycle after cycle, making it invaluable in applications like sound waves, light waves, and oscillatory movements, where patterns repeat in regular intervals.

• Understanding this property prevents mistakes when solving equations involving multiple rotations, like spinning wheels or gears.
• It also helps in converting angles beyond the typical trigonometric circle into their basic form by reducing them within the \(0\) to \(2\pi\) range.

Cosine of Angle Identities

Cosine identities like \( \cos \theta = \cos(-\theta) \) and \( \cos(\theta + 2\pi) = \cos \theta \) are essential tools in trigonometry. These help identify equivalent angles and simplify complex expressions so that calculations become straightforward.
Here are some useful points about cosine angle identities:

• The property \( \cos \theta = \cos(-\theta) \) arises from the fact the cosine is an even function.
• The property \( \cos(\theta + 2\pi) = \cos \theta \) arises from the periodicity of cosine, making it repeat every \(2\pi\) units.
• However, be mindful that some mistaken identities like \( \cos \pi = -\cos \pi \) can affect problem solving and lead to incorrect conclusions. For instance, \( \cos \pi = -1 \), and thus, it cannot equal \( -(-1) \), which would be \(1\).

These identities are not only useful for basic computations but also play a crucial role in advanced topics like calculus and Fourier series, where the manipulation of angles and periods is frequent.