Problem 99 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Determine the amplitude and period of \(y=-2 \sin (5 x)\) without graphing.

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(\frac{2\pi}{5}\)

Step 1: Identify the sine function structure

The given function is of the form \[y = a \, \sin(bx)\]. Identify the constants a and b. Here, the function is \[y = -2 \, \sin(5x)\], so \(a = -2\) and \(b = 5\).

Step 2: Determine the amplitude

The amplitude of a sine function \(y = a \, \sin(bx)\) is the absolute value of \(a\). Therefore, for \(y = -2 \, \sin(5x)\), the amplitude is \(\left| -2 \right| = 2\).

Step 3: Determine the period

The period of a sine function \(y = a \, \sin(bx)\) is given by \( \frac{2 \pi}{b} \). For the given function \(y = -2 \, \sin(5x)\), \(b = 5\). Thus, the period is \( \frac{2 \pi}{5} \).

Key Concepts

Amplitude

The amplitude of a sine function describes how far the maximum and minimum points of the wave are from the central axis, which is typically where the function crosses the x-axis. Mathematically, the amplitude is the absolute value of the coefficient 'a' in the function form \(y = a \sin(bx)\). In simpler terms, amplitude tells us how 'tall' or 'deep' the wave is. For the function \(y = -2 \sin(5x)\), the amplitude is calculated as \(|-2| = 2\). Here, the negative sign in front of the 2 does not affect the amplitude; it only influences the direction of the wave, which is reflected upside down. So, regardless of whether the value of 'a' is positive or negative, the amplitude is always a positive number representing the wave's height.

Period

The period of a sine function tells us how long it takes for the wave to complete one full cycle. For the function \(y = a \sin(bx)\), the period is calculated using the formula \( \frac{2 \pi}{b} \). Here, 'b' is the coefficient of x. For the function \(y = -2 \sin(5x)\), 'b' is 5. Plugging 'b' into the formula, we get the period as \(\frac{2 \pi}{5}\). This means the wave will repeat its pattern every \( \frac{2 \pi}{5} \) units along the x-axis. Understanding the period helps us to know how frequently the wave oscillates and is very useful in analyzing periodic phenomena.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in mathematics. They are used to relate the angles of a triangle to the lengths of its sides. Sine, represented as \(\text{sin} \), specifically deals with the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle. These functions are periodic, meaning they repeat their values in regular intervals, which is why they are extensively used for modeling waves and oscillations.

Examples include:

• The height of a point on a Ferris wheel over time.
• The position of a pendulum swinging back and forth.
• Sound waves traveling through the air.

Trigonometric functions are not only crucial in geometry but also in various fields such as physics, engineering, and even economics, where periodic behavior is observed.

Sin Function

The sine function, \( \text{sin} \), is one of the primary trigonometric functions and is pivotal in trigonometry. When graphed, the sine function creates a wave that oscillates above and below the x-axis. Its standard form is \( y = \sin(x) \), which can be modified by changing its amplitude, period, and phase. For example, in the function \(y = -2 \sin(5x)\), both the amplitude and the period are altered from the standard sine function.

Some key aspects of the sine function include:

• The function is periodic with a basic period of \(2 \pi \).
• Its maximum value is 1 and minimum is -1 in its standard form.
• It is used to model many real-world phenomena like sound and light waves.

Understanding the sine function is essential for studying wave behavior and harmonic motion in various scientific domains, making it a crucial tool in mathematical analysis.