Problem 38 - Algebra 2

Question

How many cycles does each sine function have in the interval from 0 to 2\(\pi ?\) Find the amplitude and period of each function. $$ y=-5 \sin \theta $$

Step-by-Step Solution

Answer

The function has 1 cycle in the interval from 0 to 2\(\pi\). The amplitude is 5. The period is 2\(\pi\).

Step 1: Determine the Number of Cycles

For the function \(y=-5 \sin \theta\), we need to determine the number of cycles in the interval from 0 to 2\(\pi\). A sine function completes a full cycle in every 2\(\pi\) interval. Therefore, in this case, there is 1 cycle in the given interval.

Step 2: Find the Amplitude

The amplitude is the absolute value of the coefficient of the sine function, which just means the number in front of the function. So for the given function \(y=-5 \sin \theta\), the amplitude is abs(-5) = 5.

Step 3: Determine the Period

The period of a function is the distance required for the function to complete one full cycle. For sine functions, this is typically 2\(\pi\). In this function, there is no modification to \(\theta\) inside the sine function, so the period is also 2\(\pi\).

Key Concepts

Amplitude in Sine Functions

In trigonometric functions, the amplitude is a crucial concept. Amplitude indicates the peak value or maximum displacement from the central axis of the graph of the function. For the sine function, this value tells us how far up or down the wave will reach.

• The amplitude is always a positive number.
• It is determined by the absolute value of the coefficient in front of the sine function.

For instance, in the given function \(y = -5 \sin \theta\), the coefficient of the sine function is -5.
The amplitude is the absolute value, which is \(|-5| = 5\).
Whether the sign is positive or negative, it doesn't affect the amplitude; rather, the sign impacts the direction of the wave's peaks and troughs on the graph.

Period of a Sine Function

The period of a trigonometric function describes the interval length required for the function to complete one full cycle of its pattern. In the context of the sine function, this pattern repeating interval is inherently set at \(2\pi\) for the basic sine curve.
Here are key points about the period:

• The standard cycle of the sine function takes \(2\pi\).
• If the function includes a horizontal change term (like a multiplier or addition/subtraction inside the function), this can modify the period.

In the function \(y = -5 \sin \theta\), there is no alteration in the term \(\theta\), such as a multiplier. Consequently, the period remains \(2\pi\).
Thus, the graphical representation from 0 to \(2\pi\) captures precisely one complete wave or cycle of the sine function.

Understanding Trigonometric Functions

Trigonometric functions like sine are fundamental in understanding periodic phenomena. These functions are widely applicable in fields ranging from physics to engineering.
Sine, cosine, and tangent are essential trigonometric functions each with unique features and uses.

• The sine function, \(\sin(\theta)\), relates to the y-coordinate of a point on a circle.
• The standard sine wave repeats every \(2\pi\) radians, and its sequence follows a smooth, continuous motion.

These functions can be adjusted through modifications, such as by changing amplitude or period to fit specific contexts or data patterns.
Understanding these properties aids in recognizing the nature and behavior of waves, oscillations, and cyclic events in the real world.