Problem 55 - Precalculus with Calculus Previews

Question

Graphically verify the given identity. $$ \cos (x+\pi)=-\cos x $$

Step-by-Step Solution

Answer

The identity is verified as the graph of \( y = \cos(x + \pi) \) is the negative of \( y = \cos x \).

Step 1: Understanding the Identity

The identity given is \( \cos(x + \pi) = -\cos x \). It suggests that the cosine of an angle \( x \) plus \( \pi \) is the negative cosine of \( x \). To verify this graphically, we need to understand how cosine behaves on the unit circle and its graph for transformation.

Step 2: Plotting Function Graphs

First, graph the function \( y = \cos x \) over the domain. This function oscillates between -1 and 1 with a period of \( 2\pi \). Next, graph the function \( y = \cos(x + \pi) \), which is a phase-shift of \( y = \cos x \) by \( \pi \) units to the left.

Step 3: Identifying Key Points

Examine key points where \( x = 0, \pi, 2\pi \). For \( \cos x \): \( \cos(0) = 1 \), \( \cos(\pi) = -1 \), and \( \cos(2\pi) = 1 \). When shifted by \( \pi \): \( \cos(\pi) = -1 \), \( \cos(2\pi) = 1 \), \( \cos(3\pi) = -1 \). Observing these points helps verify that the graph \( y = \cos(x + \pi) \) is actually the inverse of \( y = \cos x \).

Step 4: Comparing the Graphs

Graphically, \( y = \cos(x+\pi) \) should be a reflection of the graph \( y = \cos x \) about the x-axis. This can be checked by observing that every point on \( y = \cos(x+\pi) \) is the negative of the corresponding point on \( y = \cos x \).

Step 5: Conclusion Derived from the Graphs

Since the graph of \( y = \cos(x + \pi) \) is indeed a reflection of the graph \( y = \cos x \) over the x-axis, this verifies that the identity \( \cos(x + \pi) = -\cos x \) holds graphically.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos(x) \, \), is a fundamental trigonometric function essential in understanding wave-like patterns and oscillations. This function is often represented on the unit circle, where the x-coordinate of a point on the edge of the circle defines its value.
\( \cos(x) \, \) ranges from -1 to 1 and oscillates in this interval throughout its domain. It is periodic with a period length of \( 2\pi \, \), which means that its pattern repeats every \( 2\pi \, \) radians.
This periodicity is a key attribute in solving trigonometric identities and equations.

• Value at Key Angles: At \( x = 0 \, \), the cosine is 1; at \( x = \pi \, \), it is -1; and at \( x = 2\pi \,\), it returns to 1.
• Symmetry: \( \cos(x) \,\) is an even function. This means \( \cos(-x) = \cos(x) \,\), exemplifying its symmetrical nature about the y-axis.

Understanding the basic properties and behavior of the cosine function is fundamental when exploring more complex trigonometric identities.

Graph Transformations

Graph transformations involve moving or changing the appearance of a graph in a coordinate plane. These transformations help us understand changes in function behavior, such as shifts, stretches, and reflections.
When exploring the identity \( \cos(x+\pi)= -\cos x \,\), it involves understanding specific transformations.

• Phase Shift: The shift in \( y = \cos(x + \pi) \,\) is a horizontal translation by \( \pi \,\) units to the left. Such phase shifts do not alter the shape, only the position of the wave-like graph.
• Reflection: \( -\cos(x) \,\) indicates a reflection over the x-axis. Every point \( y \,\) for \( \cos(x) \,\) is transformed to \( -y \,\), creating an inverted image vertically.

Applying these transformations makes it easy to visualize how identities such as \( \cos(x+\pi) = -\cos x \,\) are not mere algebraic expressions but have graphical equivalents.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of the coordinate plane. It provides a geometric way to understand and visualize trigonometric functions.
For the cosine function, the unit circle helps illustrate how the angle \( x \,\) correlates to the x-coordinate of points along the circumference.

• Visualizing Cosine: On the unit circle, \( \cos(x) \,\) represents the horizontal coordinate of a point. Angles are typically measured from the positive x-axis, counterclockwise.
• Impact of \( \pi \, \): Adding \( \pi \,\) to an angle essentially rotates the point to the opposite side of the circle. This results in the x-coordinate being inverted, indicating \( \-\cos(x) \,\).

The unit circle simplifies understanding why transformations like \( \cos(x + \pi) = -\cos x \,\) hold, offering a simple visual proof for trigonometric concepts.