Problem 84 - Precalculus Essentials

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 the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin (x+\pi)=\sin x$$

Step-by-Step Solution

Answer

The graphs of \(y = \sin(x + \pi)\) and \(y = \sin(x)\) do not coincide, indicating that the equation is not an identity. A value of \(x = \pi/2\) provides an instance where both sides of the equation are defined but not equal, confirming that the equation is not an identity.

Step 1: Graph the functions

Graph the two sides of the equation, \(y = \sin(x + \pi)\) and \(y = \sin(x)\), in the same viewing rectangle. Observe whether the graphs coincide or not.

Step 2: Analyze the graphs

If the graphs coincide then the equation is an identity. If the graphs do not coincide, this indicates that the equation is not an identity. In this case, it can be observed that the two graphs do not coincide.

Step 3: Find a particular value of x

It's required to find a value of \(x\) where both sides of the equation are defined but not equal to further prove the equation is not an identity. One way to do this is to start with a value for \(x\), like 0, and calculate the results for both sides of the equation. For \(x = 0\), \( \sin(0 + \pi) = 0 \) and \( \sin(0) = 0 \). Thus, 0 is not a suitable value. Next, try \(x = \pi/2\). For this, \( \sin(\pi/2 + \pi) = -1 \) and \( \sin(\pi/2) = 1 \). Thus, for \(x = \pi/2\), both sides are defined but not equal. So, \(x = \pi/2\) is such a value.

Key Concepts

Sin (x + π)

The trigonometric function \( \sin(x + \pi) \) is a shifted version of \( \sin(x) \) along the x-axis by \( \pi \) radians, or 180 degrees. This means that the sine of an angle plus \( \pi \) is the same as the sine of that angle shifted half a cycle along the unit circle. The key property to remember is that \( \sin \) is an odd function, implying \( \sin(-x) = -\sin(x) \). Therefore, \( \sin(x + \pi) = -\sin(x) \), as \( \pi \) is equivalent to a half-turn on the unit circle, resulting in the sine value at that angle being negated.

When graphing \( \sin(x + \pi) \) alongside \( \sin(x) \) as seen in the exercise, the two will not coincide due to this property. It shows why for some values of \( x \), like \( \pi/2 \), the outputs for \( \sin(x) \) and \( \sin(x + \pi) \) are diametrically opposed.

Verifying Trigonometric Identities

Verifying trigonometric identities is a critical skill in high school and college-level mathematics. An identity is an equation that holds true for all possible values of the variable involved. In trigonometry, common identities include the Pythagorean identities, reciprocal identities, and angle sum identities.

To verify a trigonometric identity, one can apply various algebraic manipulations such as factoring, finding common denominators, or using well-known trigonometric identities to simplify one or both sides of the equation in question. In the exercise, the graphing method is used as a visual tool to assess whether \( \sin(x + \pi) = \sin(x) \) is an identity, but algebraic confirmation is also essential—particularly when graphs are not available or practical.

Graphing Trigonometric Functions

Graphing is a powerful visual method for understanding trigonometric functions. The graphs of functions like \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\) reveal patterns and properties such as periodicity, amplitude, and phase shift. Graphing \( \sin(x) \) will show a wave-like pattern oscillating between -1 and 1, representing the sine values at different angles.

When graphing \( \sin(x + \pi) \) and \( \sin(x) \) in the same viewing rectangle, it's expected to see two sine waves that have been shifted horizontally by \( \pi \) radians. This visual comparison is an essential step in verifying whether or not two trigonometric expressions form an identity, as seen in the original exercise.

Trigonometric Equations

Trigonometric equations involve unknown angles and utilize trigonometric functions and identities to solve for these angles. These equations can have multiple solutions, often an infinite number, due to the periodic nature of trigonometric functions. A typical approach to solving these equations involves isolating the trigonometric function and using inverse trigonometric functions to find the angle solutions.

If an equation is not an identity, as shown in the exercise, certain values of \( x \) will satisfy the equation while others will not. Finding a particular solution, like \( x = \pi/2 \) in the provided exercise, can demonstrate that the equation does not hold for all \( x \) and is therefore not an identity. This distinction between trigonometric equations and identities is crucial for understanding and solving such problems.