Problem 54 - Algebra and Trigonometry

Question

Adding an Echo A digital delay-device echoes an input signal by repeating it a fixed length of time after it is received. If such a device receives the pure note \(f_{1}(t)=5 \sin t\) and echoes the pure note \(f_{2}(t)=5 \cos t,\) then the combined sound is \(f(t)=f_{1}(t)+f_{2}(t)\) (a) Graph \(y=f(t)\) and observe that the graph has the form of a sine curve \(y=k \sin (t+\phi)\) (b) Find \(k\) and \(\phi\)

Step-by-Step Solution

Answer

The combined function is \( f(t) = 5\sqrt{2} \sin(t + \frac{\pi}{4}) \) with \( k = 5\sqrt{2} \) and \( \phi = \frac{\pi}{4} \).

Step 1: Set up the problem

We have two functions, \( f_1(t) = 5 \sin t \) and \( f_2(t) = 5 \cos t \), and their sum is given as \( f(t) = f_1(t) + f_2(t) \). We need to express this sum as a single sine function of the form \( y = k \sin(t + \phi) \).

Step 2: Use Sum to Product Identities

Recall that we can express \( a \sin t + b \cos t \) as \( R \sin(t + \phi) \), where \( R = \sqrt{a^2 + b^2} \). For our problem, \( a = 5 \) and \( b = 5 \) from \( f_1(t) \) and \( f_2(t) \).

Step 3: Calculate \( k \)

Compute \( R = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \). This \( R \) is our \( k \) in the sine function.

Step 4: Find \( \phi \)

We know that \( \phi \) is given by \( \tan \phi = \frac{b}{a} \), in our case \( \tan \phi = \frac{5}{5} = 1 \). This implies \( \phi = \frac{\pi}{4} \) because \( \tan(\frac{\pi}{4}) = 1 \).

Step 5: Write the combined function

Finally, write the combined function as \( f(t) = 5\sqrt{2} \sin(t + \frac{\pi}{4}) \). This matches the form \( y = k \sin(t + \phi) \) where \( k = 5\sqrt{2} \) and \( \phi = \frac{\pi}{4} \).

Key Concepts

Sine Function

The sine function, often represented as \( \sin \), is a fundamental concept in trigonometry. It is a periodic function that describes the periodic oscillations found in various natural phenomena, from light waves to sound.The basic sine function is defined on the unit circle for an angle \( t \) as the y-coordinate of the point where the terminal side of the angle intersects the unit circle. This means for an angle \( t \), the sine value is the vertical distance from the x-axis.

• The sine function is periodic, with a period of \( 2\pi \), meaning it repeats its values every \( 2\pi \) units.
• It ranges in value from -1 to 1.
• The sine of -angle is the negative of the sine of the angle, i.e., \( \sin(-t) = -\sin(t) \).

Understanding how the sine function behaves fundamentally allows us to manipulate trigonometric identities, which are crucial for solving problems like finding the coefficients and phase shifts in periodic functions.

Sum to Product Identities

Sum to product identities are trigonometric identities used to simplify expressions or solve equations involving sine and cosine terms. They transform a sum or difference of sine or cosine functions into a product, which can sometimes simplify complex trigonometric expressions.For example, the identity \( a \sin t + b \cos t = R \sin(t + \phi) \) is applied in our exercise. Here:

• \( a \) and \( b \) are amplitudes of the sine and cosine waves, respectively.
• \( R \), the single amplitude, is calculated as \( \sqrt{a^2 + b^2} \).
• \( \phi \), the phase shift, can be found using \( \tan \phi = \frac{b}{a} \).

These identities are immensely useful when trying to express the sum of trigonometric functions as a single trigonometric function, particularly in sound waves or signal processing, as seen in the given exercise.

Angle Addition

Angle addition is another important concept in trigonometry that comes into play when dealing with expressions involving different angles. The angle addition formulas allow us to find the sine or cosine of a sum or difference of angles.Consider the angle addition for sine: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]And similarly, for cosine:\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]These formulas are instrumental when we need to decompose a compound angle into simpler components or combine angles in a single sine or cosine function. By using transformations of angles, we can effectively rewrite composite functions as simpler functions, which is essential in signal processing and electrical engineering.In the exercise, such transformations helped in expressing the combined sound wave \( y = k \sin(t + \phi) \), reflecting the phase shift that's inherent when two sinusoidal functions are combined.