Problem 60 - Algebra and Trigonometry

Question

(a) Express the function in terms of sine only. (b) Graph the function. $$ f(x)=\sin x+\cos x $$

Step-by-Step Solution

Answer

(a) \( f(x) = \sqrt{2} \cos(x - \frac{\pi}{4}) \). (b) Graph is a cosine wave, shifted right by \( \frac{\pi}{4} \) with amplitude \( \sqrt{2} \).

Step 1: Use Trigonometric Identity

To express the function \( f(x) = \sin x + \cos x \) in terms of sine only, we can use the trigonometric identity that \( \cos x = \sin(\frac{\pi}{2} - x) \). Therefore, we write: \[ \cos x = \sin\left(\frac{\pi}{2} - x\right) \]

Step 2: Substitute Identity into Function

Substitute the identity from step 1 into the original function: \[ f(x) = \sin x + \cos x = \sin x + \sin\left(\frac{\pi}{2} - x\right) \] Now, we have expressed \( f(x) \) using sine functions only.

Step 3: Simplify Function Further

Using the sine addition formula \( \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \), set \( A = x \) and \( B = \frac{\pi}{2} - x \): \[ f(x) = 2 \sin\left(\frac{x + (\frac{\pi}{2} - x)}{2}\right) \cos\left(\frac{x - (\frac{\pi}{2} - x)}{2}\right) \] Simplifying inside the sine function: \( \frac{x + \frac{\pi}{2} - x}{2} = \frac{\pi}{4} \) and inside the cosine function: \( \frac{x - \frac{\pi}{2} + x}{2} = x - \frac{\pi}{4} \). Thus, the function is: \[ f(x) = 2 \sin\left(\frac{\pi}{4}\right) \cos\left(x - \frac{\pi}{4}\right) \]

Step 4: Express Using Sine with Simplification

Knowing that \( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \), we further simplify: \[ f(x) = \sqrt{2} \cos\left(x - \frac{\pi}{4}\right) \] Although it looks like cosine, the amplitude is consistent with a transformation from the sine function because this follows from the sum identity representation.

Step 5: Graph the Function

Graph the function \( f(x) = \sqrt{2} \cos\left(x - \frac{\pi}{4}\right) \). This is a cosine wave shifted to the right by \( \frac{\pi}{4} \) and has an amplitude of \( \sqrt{2} \). It wavers between \( -\sqrt{2} \) and \( \sqrt{2} \). To confirm correctness, overlay \( \sin x + \cos x \) and see that they match exactly.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions that describes a smooth, periodic oscillation. It is generally expressed as \( \sin x \), where \( x \) represents the angle. This function exhibits a wave-like behavior, undulating between -1 and 1. The sine wave is notable for its symmetry and periodicity, repeating every \( 2\pi \) radians.

• Period: The function has a period of \( 2\pi \), meaning the wave pattern repeats every \( 2\pi \) units.
• Amplitude: This is the peak value from the center of the wave, typically 1 for the standard sine function.
• Key Characteristics: The function intersects the x-axis at multiples of \( \pi \), such as 0, \( \pi \), and \( 2\pi \).

Sine functions are at the heart of modeling wave-like phenomena such as sound waves, light waves, and tides. Understanding sine is foundational for studying more complex trigonometric transformations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variables. They serve as essential tools for simplifying expressions and solving equations. In the context of this exercise, the identity \( \cos x = \sin\left(\frac{\pi}{2} - x\right) \) is pivotal.

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \), which forms the basis for many other identities.
• Angle Sum and Difference Identities: Such as \( \sin(A + B) = \sin A \cos B + \cos A \sin B \), allowing for complex expression breakdown.
• Co-function Identities: These relate the sine and cosine of complementary angles, for example, the identity used above.

These identities are crucial because they enable the conversion of seemingly complex trigonometric expressions into simpler forms. This simplification is a primary skill in solving trigonometric equations and transforming functions.

Function Transformation

Function transformation refers to the shifting, stretching, or reflecting of a basic function to form a new function. When dealing with trigonometric functions, transformations are particularly useful in modeling time-varying phenomena.

• Vertical and Horizontal Shifts: Shifting the graph up, down, left, or right. For example, the function \( f(x) = \sqrt{2} \cos(x - \frac{\pi}{4}) \) involves a horizontal shift by \( \frac{\pi}{4} \).
• Amplitude Changes: Transformations that alter the function's amplitude affect how tall or short the wave appears. In our example, the amplitude is \( \sqrt{2} \), which stretches the regular cosine wave.
• Period Adjustments: Adjusting the period alters how frequently the wave repeats over a specified interval. Often achieved by modifying the frequency within the function.

Understanding function transformations aids in graphing trigonometric functions accurately and predicting their behavior. This conceptual grasp is key in fields ranging from physics to engineering and beyond.