Question

True or False The amplitude of \(y=\frac{1}{2} \cos x\) is \(1 .\) Justify your answer.

Short answer

False. The amplitude of \(y=\frac{1}{2} \cos x\) is \(\frac{1}{2}\), not 1.

Step-by-step solution

Identify the Coefficient of the cosine function

In the given equation \(y=\frac{1}{2} \cos x\), the coefficient of the cosine function is \(\frac{1}{2}\). The coefficient of the cosine function represents the amplitude of the function.

Compare the Amplitude with given value

Since the coefficient of the cosine function is equivalent to the amplitude, and in the given equation the coefficient of the cosine function (the amplitude of the function) is \(\frac{1}{2}\), it is concluded that the amplitude is \(\frac{1}{2}\), not 1.

Key concepts

Trigonometric Functions

Trigonometry is a branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right-angled triangles. This discipline introduces several special functions to describe these relationships: sine, cosine, and tangent, along with their reciprocals cosecant, secant, and cotangent. Each of these functions relates an angle of a triangle to a ratio of two of its sides.

The cosine function is particularly noteworthy among these because it projects the length of the adjacent side over the hypotenuse for a given angle when the triangle is right-angled. Essentially, if we consider a unit circle, where all the radii are of length one, the cosine of an angle will give us the x-coordinate of the point where the radius forms that angle with the x-axis.

Amplitude of a Cosine Function

The concept of amplitude is central to understanding the properties of the cosine function. In physics, amplitude generally refers to the maximum extent of a vibration or displacement of a wave. When transferred to trigonometry, amplitude denotes the maximum value that the function reaches, which is the 'height' of the wave from the central axis.

In simple terms for the cosine function, the amplitude represents how far the graph of the function moves above and below the horizontal axis (usually the x-axis) on a graph. For the standard cosine function, written as \( y = \text{cos}(x) \), the amplitude is 1 since the function oscillates from -1 to 1. However, when we see an equation like \( y = A \text{cos}(x) \), the constant A represents the amplitude, which rescales the wave, either stretching or compressing it vertically.

Identifying Coefficients

In trigonometry, coefficients play a crucial role in defining the characteristics of trigonometric functions. To identify the coefficients in an equation like \( y = A \text{cos}(x) \), we look for the constant term that is multiplied directly by the cosine function. This term A is the amplitude coefficient, which tells us how tall or how short the waves of our cosine graph will be.

Understanding coefficients isn't limited to amplitude; it also includes the frequency and phase shift. For instance, in \( y = A \text{cos}(Bx - C) + D \), A influences the amplitude, B affects the frequency, C shifts the graph horizontally, and D moves it vertically. But for the amplitude alone, the coefficient attached directly to \( \text{cos}(x) \) is our main focus.

Trigonometry Problems

Trigonometry problems can range from simple angle finding tasks to complex waveform analyses. When tackling problems that involve trigonometric functions, it is essential to first identify key features like amplitude, frequency, phase shift, and vertical translation.

Problems like determining the amplitude from a cosine function equation require careful examination of the coefficients. These problems are fundamental exercises to grasp the concepts of trigonometry and apply them effectively. By learning to identify these various coefficients and how they alter a trigonometric graph, students can solve a wide range of practical problems, from predicting tides in oceanography to designing electrical circuits in engineering.