Problem 73 - Precalculus

Question

For the following exercises, simplify the expression, and then graph both expressions as functions to verify the graphs are identical. $$ \sin \left(\frac{\pi}{4}+x\right) $$

Step-by-Step Solution

Answer

Simplified expression: \( \frac{\sqrt{2}}{2} (\cos x + \sin x) \). Graphs are identical.

Step 1: Use the Angle Addition Formula for Sine

To simplify the expression \( \sin \left( \frac{\pi}{4} + x \right) \), we will use the angle addition formula for sine which is \( \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \). Here, \( a = \frac{\pi}{4} \) and \( b = x \).

Step 2: Substitute Values into Formula

Substitute \( a \) and \( b \) into the angle addition formula: \[\sin \left( \frac{\pi}{4} + x \right) = \sin \left( \frac{\pi}{4} \right) \cdot \cos x + \cos \left( \frac{\pi}{4} \right) \cdot \sin x\] Since \( \sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \), substitute these values in.

Step 3: Simplify the Expression

Replacing \( \sin \left( \frac{\pi}{4} \right) \) and \( \cos \left( \frac{\pi}{4} \right) \) with \( \frac{\sqrt{2}}{2} \), the expression becomes:\[\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x = \frac{\sqrt{2}}{2} (\cos x + \sin x)\] This is the simplified expression.

Step 4: Graph the Functions

To verify that the graphs are identical, plot both the original function \( \sin \left( \frac{\pi}{4} + x \right) \) and the simplified function \( \frac{\sqrt{2}}{2} (\cos x + \sin x) \). Check that both graphs overlap perfectly over the same interval, demonstrating they represent the same function.

Key Concepts

Understanding the Sine Function

The sine function is one of the fundamental trigonometric functions, often represented as \( \sin(x) \). It defines the y-coordinate of a point on a unit circle as it corresponds to a given angle \( x \). This function is periodic, which means it repeats its values in regular intervals, specifically every \( 2\pi \). The sine function smoothly oscillates between -1 and 1, creating a wave-like graph, known as a sinusoidal curve. The understanding of sine is crucial in solving trigonometric equations and simplifying expressions using identities like the angle addition formula. By learning how \( \sin \left( a + b \right) \) breaks into components using sine and cosine, you can work with various trigonometric problems easily. Also, values such as \( \sin(\frac{\pi}{4}) \) are standard, resulting in \( \frac{\sqrt{2}}{2} \), a helpful value to remember for solving problems quickly.

Exploring Trigonometric Identities

Trigonometric identities are essential tools in simplifying and solving equations involving trigonometric functions like sine, cosine, and tangent. These identities are equations that hold true for all values of the variables involved. One of the key identities used in trigonometry is the angle addition formula.

• The Angle Addition Formula for sine is: \( \sin(a + b) = \sin a \cdot \cos b + \cos a \cdot \sin b \)

This formula helps unravel expressions where two angles are added together, simplifying them into basic sine and cosine terms. For example, applying the formula to \( \sin \left( \frac{\pi}{4} + x \right) \) allows us to transform it into simpler parts that make graphing and further calculations easier.Using identities is not just about breaking down expressions; it's about recognizing patterns that can be essential in understanding complex mathematical concepts. Getting comfortable with these identities can significantly ease the process of verifying function equality and proving mathematical theorems.

Graphing Functions

Graphing trigonometric functions like sine and cosine involves plotting points based on angle inputs and corresponding function outputs. When you graph functions, it's important to know the function's period, amplitude, and phase shift. These characteristics influence the wave's shape and positioning on a coordinate plane.To graph the function \( \sin(\frac{\pi}{4} + x) \), you should think about:

• **Phase Shift**: The function \( \frac{\pi}{4} + x \) indicates a leftward shift from \( \sin(x) \)
• **Amplitude**: The maximum value remains unchanged as sine's inherent range is from -1 to 1 unless altered by a coefficient
• **Periodicity**: Still repeats every \( 2\pi \)

When graphed, the simplified expression \( \frac{\sqrt{2}}{2} (\cos x + \sin x) \) should create an identical curve, thus verifying that both expressions describe the same function. Overlaying both graphs verifies their equivalence through perfect alignment. Exploring these graphs helps visualize the relationship between various trigonometric expressions, aiding in deeper comprehension beyond mere calculation.