Problem 11 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=-4 \sin x $$

Step-by-Step Solution

Answer

The amplitude is 4 and the period is \(2\pi\).

Step 1: Determine the Amplitude

The amplitude of a sine or cosine function is the absolute value of the coefficient of sine or cosine. Here, take the absolute value of -4, which is 4.

Step 2: Determine the Period

The period of the sine function is \(2\pi\) divided by the absolute value of the coefficient of x. In this case, there is no specific coefficient for x. Therefore, the period remains the default period, which is \(2\pi\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function noted for its wavelike pattern. It forms the basis of many applications in mathematics, especially in the study of periodic phenomena such as sound waves, light waves, and tides. The general form of a sine function is expressed as:\[ y = A \sin(Bx + C) + D \]where:- \(A\) is the amplitude,- \(B\) determines the period,- \(C\) accounts for horizontal shifts,- \(D\) represents vertical shifts.

This function oscillates between its maximum and minimum values, creating a smooth, continuous wave. Its graph is sinusoidal, meaning it has a repetitive pattern that repeats every full cycle.

Trigonometric Functions

Trigonometric functions are crucial for studying relationships in right-angled triangles and modeling periodic phenomena. Apart from sine, other primary trigonometric functions include cosine and tangent. Each function has unique properties:

• **Sine (\( \sin \))**: This function gives the ratio of the opposite side to the hypotenuse in a right triangle.
• **Cosine (\( \cos \))**: It represents the ratio of the adjacent side to the hypotenuse.
• **Tangent (\( \tan \))**: This function is the ratio of the sine to the cosine, or opposite over adjacent.

All these functions are periodic, meaning they repeat their values in regular intervals. This makes them highly valuable in analyzing cycles and patterns in various fields such as physics and engineering.

Understanding these functions and their behaviors is essential for solving complex problems in mathematics.

Amplitude

The amplitude of a trigonometric function, like the sine function, refers to the height from the middle line of the wave to its peak or trough. In the function \(y = -4 \sin x\), the high point (crest) and low point (trough) each rely on the amplitude. The absolute value of the coefficient before the sine function, which is \(-4\) in this case, gives us the amplitude.

Thus, the amplitude here is \(|-4|\ = 4\.\) This means the graph of the function will range from -4 to 4, moving equally above and below the x-axis. A crucial aspect of trigonometric functions is their amplitudes' indication of the wave’s strength or intensity.

Period of a Function

The period of a trigonometric function is the duration it takes for the function to repeat its pattern. For the basic sine function \( \sin x \), the period is \(2\pi\). This means after each interval of \(2\pi\), the function starts a new cycle.

When altered, as with a coefficient before the \(x\) value, the period can change significantly. However, in the case of \(-4 \sin x\), there is no coefficient to modify the \( x \), so the period remains \(2\pi\), its default length. The concept of the period is a cornerstone in understanding cycles, helping in predicting the timing of peaks and troughs.