The study of trigonometric functions often involves understanding the periodicity and symmetry of these functions. A key element in this understanding lies in recognizing the relationship between trigonometric values at different angles. One common observation is the pattern of angles that lead to equivalent trigonometric values. This pattern is often expressed in the form of general solutions for trigonometric equations. In this article, we will explore the relationship between the expressions $n\pi +(-1)^n \pi/6$ and $n\pi + \pi/6$, where $n$ is an integer, and how this relates to the general solutions of trigonometric equations involving sine and cosine.
Understanding the Pattern
To understand the relationship between these two expressions, let's examine the behavior of the term $(-1)^n$. When $n$ is an even integer, $(-1)^n$ equals 1, and when $n$ is odd, $(-1)^n$ equals -1. This means:
- For even values of n: $n\pi +(-1)^n \pi/6 = n\pi + \pi/6$
- For odd values of n: $n\pi +(-1)^n \pi/6 = n\pi - \pi/6$
This pattern highlights that the expression $n\pi +(-1)^n \pi/6$ generates two distinct sets of angles:
- One set: $n\pi + \pi/6$ where $n$ is an even integer.
- Another set: $n\pi - \pi/6$ where $n$ is an odd integer.
Let's illustrate this with a few examples:
- n = 0 (even): $0\pi + (-1)^0 \pi/6 = \pi/6$
- n = 1 (odd): $1\pi + (-1)^1 \pi/6 = \pi - \pi/6 = 5\pi/6$
- n = 2 (even): $2\pi + (-1)^2 \pi/6 = 2\pi + \pi/6$
- n = 3 (odd): $3\pi + (-1)^3 \pi/6 = 3\pi - \pi/6 = 17\pi/6$
The Connection to General Solutions
The expressions we have explored are crucial in understanding the general solutions of trigonometric equations involving sine and cosine. Recall that the sine function has a period of $2\pi$ and is symmetric about the origin. This means that the sine function takes on the same value at angles that differ by multiples of $2\pi$. Similarly, the cosine function also has a period of $2\pi$ but is symmetric about the y-axis. Therefore, the cosine function takes on the same value at angles that differ by multiples of $2\pi$ but also exhibit a symmetry property.
Sine Function:
If $\sin \theta = a$, then the general solution for $\theta$ can be expressed as:
$\theta = n\pi + (-1)^n \alpha$, where $\alpha$ is the principal solution (an angle between $0$ and $\pi$) and $n$ is an integer.
Notice that the expression $n\pi + (-1)^n \alpha$ encompasses both the periodicity and the symmetry of the sine function. The term $n\pi$ accounts for the periodicity, and the term $(-1)^n \alpha$ ensures that the solution covers all possible angles, considering the symmetry of the function.
Cosine Function:
If $\cos \theta = b$, then the general solution for $\theta$ can be expressed as:
$\theta = 2n\pi \pm \beta$, where $\beta$ is the principal solution (an angle between $0$ and $\pi$) and $n$ is an integer.
Here, the $2n\pi$ accounts for the periodicity of the cosine function, and the $\pm \beta$ term incorporates the symmetry about the y-axis.
An Example of Application
Let's consider the equation $\sin \theta = \frac{1}{2}$. The principal solution is $\theta = \frac{\pi}{6}$. Applying the general solution formula for sine functions, we have:
$\theta = n\pi + (-1)^n \frac{\pi}{6}$
This formula captures all the possible values of $\theta$ that satisfy the given equation. For example, when $n=0$, we get $\theta = \frac{\pi}{6}$, and when $n=1$, we get $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$.
Conclusion
In conclusion, the expressions $n\pi +(-1)^n \pi/6$ and $n\pi + \pi/6$ are not equivalent, but they are closely related due to the behavior of the term $(-1)^n$. This relationship is crucial in understanding the general solutions of trigonometric equations involving sine and cosine. These general solutions encapsulate the periodicity and symmetry of the trigonometric functions, ensuring that all possible solutions are captured. By understanding these relationships and applying them to trigonometric equations, we gain a deeper insight into the behavior of trigonometric functions and their role in various mathematical applications.
