Problem 51 - Contemporary Precalculus

Question

Express \(\cos 3 x\) in terms of \(\cos x\)

Step-by-Step Solution

Answer

Answer: \(\cos 3x = 4\cos^3 x - 3\cos x\)

Step 1: Use the triple angle formula for cosine

The triple angle formula for cosine is given by: $$\cos 3x = 4\cos^3 x - 3\cos x$$

Step 2: Write the final expression

Since there are no \(\sin x\) terms present in the formula, the expression for \(\cos 3x\) in terms of \(\cos x\) is: $$\cos 3x = 4\cos^3 x - 3\cos x$$

Key Concepts

Trigonometry Basics

Trigonometry is a fundamental branch of mathematics that deals with the relationships between the angles and sides of triangles. It is used in various fields such as engineering, physics, and even computer science. The most common trigonometric functions are sine, cosine, and tangent. Each function has a special relationship with an angle in a right triangle.
The core idea is to understand how these functions relate the measurements of angles to the lengths of sides in triangles. Trigonometry relies heavily on algebra and geometry, helping us extend the concepts of measurements and calculations.

• Sine (\( ext{sin}\)): This function relates the opposite side to the hypotenuse of a right triangle.
• Cosine (\( ext{cos}\)): It connects the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)): Tangent is concerned with the relationship between the opposite side and the adjacent side of a right triangle.

Understanding these relationships allows for solving complex problems, such as determining the height of a building or the distance across a river, just with some measurements and angle calculations.

Understanding the Cosine Function

The cosine function always deals with the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle. In mathematical terms, it is expressed as:\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]This function is integral to trigonometry and is widely used in numerous applications. The cosine of an angle is a measure of how far the angle "leans" from being a full vertical.Additionally, the cosine function is not limited to geometrical interpretations. It's also part of the periodic functions, repeating its values over a set interval, specifically every \(2\pi\) radians. This periodic nature makes it useful in modeling cyclical patterns, like sound waves, light waves, and even the changing seasons.
In trigonometric identities, cosine frequently appears and interacts with other functions, like sine and tangent, forming foundational identities that facilitate the solving of many trigonometric equations.

Exploring Mathematical Identities

Mathematical identities are equations that hold true for all values of the variables involved. In trigonometry, identities are used to simplify expressions and solve equations more efficiently. They provide relationships between trigonometric functions and allow the transformation of complex trig expressions into simpler forms.Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum and Difference Identities: Useful for expressing functions like \(\cos(a + b)\)
• Double Angle Identities: These relate \(\sin 2x, \cos 2x\) with \(\sin x, \cos x\)

In the context of the given problem, the Triple Angle Formula is used: \[ \cos 3x = 4\cos^3 x - 3\cos x \]This formula illustrates an identity for transforming expressions of multiples of angles, finding particular use in calculus and physics due to its utility in handling periodic and wave functions.By mastering these identities, students can more easily navigate through complex trigonometric problems and appreciate the interconnected nature of these mathematical principles. They are essential tools in any mathematician's toolkit.