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Chapter 20: Problem 46

Solve the given problems. Express \(\cos 3 x\) in terms of \(\cos x\) only.

Short Answer

Expert verified
\( \cos 3x = 4\cos^3 x - 3\cos x \).

Step by step solution

01

Understanding the Triple Angle Formula

The goal is to express \( \cos 3x \) in terms of \( \cos x \). We use the triple angle formula for cosine, which states: \[ \cos 3x = 4\cos^3 x - 3\cos x \] This formula allows us to directly express \( \cos 3x \) using only \( \cos x \).
02

Substituting into the Formula

To find \( \cos 3x \), simply substitute \( \cos x \) into the formula we identified: \[ \cos 3x = 4\cos^3 x - 3\cos x. \] No further steps are needed because the entire transformation has been done using the triple angle formula.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Functions
Cosine functions are one of the building blocks of trigonometry. They help us understand angles and cycles, which appear in various mathematical and physical problems. Cosine, specifically, is a function that relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse.
  • The cosine function is periodic with a period of \(2\pi\), meaning it repeats its values in a regular cycle.
  • It ranges from -1 to 1 and is known for its smooth waving graph, reminiscent of rolling waves.
  • This function is key when dealing with waves, circular motion, or analyzing the behavior of oscillations.
By understanding the basics of cosine functions, you can dive deeper into more advanced topics like trigonometric identities and transformations. Here, the cosine triple angle formula is an extension of the basic cosine concept.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for any value of the variable where both sides of the equation are defined. These identities are crucial tools in simplifying expressions and solving trigonometric equations.
  • One of the primary identities is the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\).
  • There are sum and difference identities, like \(\cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b\).
  • Then we have angle multiples, like double and triple angle identities. The triple angle formula, \(\cos 3x = 4\cos^3 x - 3\cos x\), is one such identity.
These identities are incredibly useful because they allow us to transform complex expressions into simpler forms. This ability to reduce complexity helps us solve equations that would otherwise be difficult to manage. Understanding how to apply these identities is key to mastering trigonometry.
Mathematical Transformations
Mathematical transformations involve changing a function or an equation into a different but equivalent form, allowing for easier interpretation and solution. They are vital in simplifying expressions, especially in trigonometry.
  • When applying transformations to trigonometric functions, it often involves using identities to rewrite expressions.
  • For example, in transforming \(\cos 3x\) into a formula involving only \(\cos x\), we lever the triple angle identity: \(\cos 3x = 4\cos^3 x - 3\cos x\).
  • Transformations can also change the graph of a function in terms of shifts, stretches, and reflections, helping us understand how the function behaves in different contexts.
By employing these transformations, equations and functions become more manageable, facilitating the resolution of complex mathematical problems. They unlock new perspectives and solutions, proving to be an essential aspect of mathematical problem-solving.

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Most popular questions from this chapter

Solve the given problems. In the analysis of the angles of incidence \(i\) and reflection \(r\) of a light ray subject to certain conditions, the following expression is found: \(E_{2}\left(\frac{\tan r}{\tan i}+1\right)=E_{1}\left(\frac{\tan r}{\tan i}-1\right)\) Show that \(E_{2}=E_{1} \frac{\sin (r-i)}{\sin (r+i)}\)

Without graphing, determine the amplitude and period of the function \(y=4 \sin x \cos x .\) Explain.

Solve the given equations graphically. An equation used in astronomy is \(\theta-e \sin \theta=M .\) Solve for \(\theta\) for \(e=0.25\) and \(M=0.75\)

Use the given substitutions to show that the given equations are valid. In each, \(0<\theta<\pi / 2\). $$\text { If } x=\cos \theta, \text { show that } \sqrt{1-x^{2}}=\sin \theta$$

Solve the given trigonometric equations analytically and by use of a calculator. Compare results. Use values of \(x\) for \(0 \leq x<2 \pi\). $$2 \sin x=\tan x$$
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