Problem 8 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

\text { Write a formula for } \cos 3 x \text { entirely in terms of sums and powers of } \cos x \text { . }

Step-by-Step Solution

Answer

The formula \( \cos 3x \) in terms of \( \cos x \) is \( 4\cos^3x - 3\cos x \).

Step 1: Remember the triple angle identity for cosine

The identity for \( \cos 3x \) is \( \cos 3x = 4\cos^3x - 3\cos x\), which is a standard trigonometric identity.

Step 2: Express \( \cos 3x \) in terms of \( \cos x \)

Replace \( \cos 3x \) in the given equation with \( 4\cos^3x - 3\cos x \). No further manipulation is required because it's already in terms of \( \cos x \). Therefore, the expression for \( \cos 3x \) in terms of \( \cos x \) is \( 4\cos^3x - 3\cos x \).

Key Concepts

Cosine

The cosine function is one of the fundamental trigonometric functions, often abbreviated as 'cos.' It's essential in the field of trigonometry, which deals with the relationships between the angles and sides of triangles, particularly right-angled triangles.

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. Represented symbolically for an angle θ, this ratio is written as \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

Furthermore, the cosine function has a range of applications in physics and engineering, especially in the analysis of waves and oscillations. For instance, in terms of a unit circle where the radius is 1, the cosine value represents the x-coordinate of a point on the circle corresponding to a particular angle.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold true for all values of the variables where both sides of the identity are defined. These identities are crucial for simplifying and solving trigonometry problems. Some of the most widely used are the Pythagorean identities, angle sum and difference identities, double angle identities, and the subject of our interest — the triple angle identities.

Triple angle identities express functions of triple angles in terms of functions of simple angles. They are particularly handy in calculus and complex problem-solving tasks even beyond the scope of geometry, such as in Fourier series in mathematics and signal processing.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It's particularly known for its rules and formulas which are widely used in various scientific fields, including physics, engineering, astronomy, and geography.

In trigonometry, we work with a variety of functions including sine, cosine, tangent, cotangent, secant, and cosecant. The functions commonly appear in a series of mathematical problems and real-life applications like the calculation of heights and distances, the design of components in mechanical systems, and even in the analysis of sound waves.

Function Transformation

Function transformation involves the alteration of a parent function's graph to produce a new graph. This can be achieved through various operations like shifting, reflecting, stretching, or compressing. In trigonometry, transformations are particularly important when dealing with trigonometric functions which repeat over a specific interval, known as the period.

For example, multiplying the input of a cosine function by a constant will affect its period, effectively compressing or stretching the graphical representation of the function. This principle is exhibited in the triple angle formula, where we transform \( \cos x \) into \( \cos 3x \) showing a function that oscillates three times as fast as the original cosine function. Understanding how to manipulate and transform these functions is crucial for grasping more complex concepts and their practical applications.