Problem 90 - Algebra and Trigonometry

Question

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \sin \left(x+\frac{\pi}{2}\right)=\sin x+\sin \frac{\pi}{2} $$

Step-by-Step Solution

Answer

The equation \(\sin \left(x+\frac{\pi}{2}\right)=\sin x+\sin \frac{\pi}{2}\) is not an identity as its graphs do not coincide for all \(x \). One such \(x\) value is \(x = 0 \), which satisfies that both sides are defined but not equal. The value is \(1\) for the right side and \(1\) for the left side.

Step 1: Understanding the equation

The given equation \(\sin \left(x+\frac{\pi}{2}\right)=\sin x+\sin \frac{\pi}{2}\) needs to be checked if it's an identity or not. That can be done by graphing both sides and seeing if they coincide. If they do, it's an identity, otherwise it's not an identity.

Step 2: Analyzing the right side of the equation

The right side of the equation is \(\sin x+\sin \frac{\pi}{2}\). The term \(\sin \frac{\pi}{2}\) is a constant which is equal to 1. So, the right side simplifies to \(\sin x + 1\). This is a standard sine curve, but shifted 1 unit upwards.

Step 3: Analyzing the left side of the equation

The left side of the equation is \(\sin \left(x+\frac{\pi}{2}\right)\). This can be seen as a standard sine curve, but shifted \(\frac{\pi}{2}\) units to the left. This is due to the fact that if \(\theta = x+\frac{\pi}{2}\), then \(\sin \theta\) is a normal sine curve.

Step 4: Graphing and visualization

If we graph both sides, we find that they do not coincide for all values of \(x \). This indicates that the given equation is not an identity.

Step 5: Finding a value for \(x\)

According to the task, we also need to find a value of \(x\) for which both sides are defined but not equal. One such value can be \(x = 0 \). For \(x=0\), the right side of the equation is \(\sin(0) + 1 = 1\), while the left side of the equation is \(\sin(0 + \frac{\pi}{2}) = \sin(\frac{\pi}{2}) = 1\). So, \(x = 0 \) is a valid solution.

Key Concepts

Sine Function

The sine function is a fundamental part of trigonometry, representing how the y-coordinate of a unit circle changes as we move around it. Consider a unit circle in which the radius is 1. As a point moves counterclockwise from the positive x-axis around the circle, the sine of the angle associated with that point is equal to the y-coordinate of the point.

Key properties of the sine function include:

• It is periodic with a period of \(2\pi\), meaning \(\sin(x) = \sin(x + 2\pi)\).
• Its range lies between -1 and 1, giving the sine function its signature wave shape.
• It is an odd function; therefore, \(\sin(-x) = -\sin(x)\).

These properties are essential when understanding how various transformations affect the graph of the sine function. The sine function is also known for being smooth and continuous, meaning there are no abrupt changes or breaks.

Graphing Equations

Graphing equations involving trigonometric functions can be a powerful visual method to understand their behaviors and relationships. When graphing, the basic sine function \(\sin x\), creates a smooth, wave-like curve that oscillates between -1 and 1.

When we need to graph changes like \(\sin(x + \frac{\pi}{2})\) versus \(\sin x + 1\), it helps us visualize how transformations affect the graph:

• Horizontal shifts, like \(\sin(x + \frac{\pi}{2})\), move the graph left or right.
• Vertical shifts, like adding a 1 in \(\sin x + 1\), move the graph up or down.

By plotting both sides of the equation \(\sin(x + \frac{\pi}{2}) = \sin x + \frac{\pi}{2}\) in a graphing window, these transformations reveal if they coincide or not. In this particular case, they do not fully overlap, indicating the original equation is not an identity.

Trigonometric Transformations

Trigonometric transformations alter the standard trigonometric functions in various ways, affecting their amplitude, period, phase shift, and vertical shift. Understanding these modifications can help us solve equations involving trigonometric identities.

Types of transformations:

• Amplitude changes: Affect the height and depth of the sine wave from its center line.
• Period changes: Alter the length required for the graph to complete one full cycle.
• Phase shifts: Like \(\sin(x + \frac{\pi}{2})\), move the wave left or right along the x-axis.
• Vertical shifts: Such as \(\sin x + 1\), translate the graph up or down.

These transformations are helpful when analyzing the behavior of trigonometric functions in different contexts and help in verifying identities, as they show how graphs change and interact.