Problem 67 - Precalculus

Question

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. $$ \cos \left(\frac{\pi}{2}-x\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(\sin(x)\), and both graphs are identical.

Step 1: Identify the Trigonometric Identity

Recognize that the expression \(\cos\left(\frac{\pi}{2}-x\right)\) can be simplified using a known trigonometric identity: the co-function identity. The co-function identity for cosine states that \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\). This means our expression simplifies directly to \(\sin(x)\).

Step 2: Simplify the Expression

Using the co-function identity from Step 1, we simplify \(\cos\left(\frac{\pi}{2}-x\right)\) to \(\sin(x)\). So, our simplified expression is \(\sin(x)\).

Step 3: Graph the Original Function

Graph the function \(f(x) = \cos\left(\frac{\pi}{2}-x\right)\). This function describes the cosine of the complement of \(x\), which, as we've simplified, behaves the same as \(\sin(x)\).

Step 4: Graph the Simplified Function

Graph the function \(g(x) = \sin(x)\). The graph of this function is a familiar sine wave, which oscillates between -1 and 1, with periods occurring every \(2\pi\) units.

Step 5: Validate by Comparison

Compare the graphs of \(f(x) = \cos\left(\frac{\pi}{2}-x\right)\) and \(g(x) = \sin(x)\). Both should be identical, confirming our simplification as they both resemble the standard sine wave shape. This validates that our simplified expression is accurate.

Key Concepts

Co-function Identities

In trigonometry, co-function identities are relationships between trigonometric functions of complementary angles. Complementary angles are two angles that add up to \(\frac{\pi}{2}\) radians (or 90 degrees). For these angles, the trigonometric identities relate sine and cosine functions, as well as tangent and cotangent, secant and cosecant. Here is the key point:

• \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\)
• \(\sin\left(\frac{\pi}{2} - x\right) = \cos(x)\)
• \(\tan\left(\frac{\pi}{2} - x\right) = \cot(x)\)

Notice how when dealing with \(\frac{\pi}{2} - x\), each trigonometric function turns into its respective co-function. These identities simplify calculations and are useful in transforming expressions for easier graphing lessons.

Cosine Function

The cosine function, denoted as \(\cos(x)\), is one of the fundamental trigonometric functions. It originates from the unit circle, which is a circle with a radius of 1. The function \(\cos(x)\) gives the x-coordinate of a point on the unit circle as angle \(x\) varies. Here are some features of the cosine function:

• The cosine curve starts at 1 when \(x = 0\).
• It oscillates between -1 and 1.
• The period of the cosine function is \(2\pi\), meaning it repeats every \(2\pi\) radians.
• It is an even function, so \(\cos(-x) = \cos(x)\).

The graph of \(\cos(\cdot)\) looks like a smooth, continuous wave that starts from a peak. Its behavior and properties make it a building block for understanding more complex trigonometric concepts.

Sine Function

The sine function, \(\sin(x)\), is another core trigonometric function. Like the cosine function, it is derived from the unit circle, and its value represents the y-coordinate at a point when moving counterclockwise around the circle by angle \(x\). Critical aspects include:

• Sine starts at 0 when \(x = 0\).
• It ranges from -1 to 1, creating a series of smooth waves.
• Its period, like cosine, is \(2\pi\).
• The sine function is odd, so \(\sin(-x) = -\sin(x)\).

One of the hallmarks of sine is its wave starting at zero, growing positively, and symmetrically moving in the opposite direction as it follows its natural rhythm across the x-axis. Its predictability is quite useful in solving equations and modeling real-world phenomena.

Graphing Functions

Graphing functions is a powerful visual tool in mathematics that helps us understand the behavior of different equations. When graphing trigonometric functions, like sine and cosine, follow these basic steps:

• Identify the function you wish to graph (e.g., \(\cos(\cdot)\) or \(\sin(\cdot)\)).
• Determine the range and period of the function. Most trigonometric functions range from -1 to 1, with a period of \(2\pi\).
• Plot key points: the starting point, maximum, minimum, and endpoints of one period.
• Draw a smooth, continuous wave connecting these points.
• Repeat the pattern to cover the desired domain.

Graphing both \(f(x) = \cos\left(\frac{\pi}{2} - x\right)\) and \(g(x) = \sin(x)\) allows you to see they are identical, validating simplifications made through co-function identities. This pictorial approach to understanding math concepts makes verifying calculations less abstract and more intuitive.