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Chapter 5: Problem 25

Rewrite \(\cos 4 x\) in terms of \(\cos x\).

Short Answer

Expert verified
\(\cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1\)

Step by step solution

01

Identify Relevant Trigonometric Identity

Recall the following trigonometric identity for \(\cos 2x\): \(\cos 2x = 2 \cos^2 x - 1\)
02

Replace x with 2x in the Trigonometric Identity

Replace \(x\) with \(2x\) in the equation \(\cos 2x = 2 \cos^2 x - 1\), to obtain the equation for \(\cos 4x\): \(\cos 4x = 2 \cos^2 2x - 1\)
03

Replace \(\cos 2x\) with its Identity

Replace \(\cos 2x\) in the equation \(\cos 4x = 2 \cos^2 2x - 1\) with its identity \(\cos 2x = 2 \cos^2 x - 1\). This gives \(\cos 4x = 2 (2 \cos^2 x - 1)^2 - 1\)
04

Simplify the Equation

Expand and simplify the equation. Result: \( \cos 4x = 8 \cos^4 x - 8 \cos^2 x + 1 \)

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Key Concepts

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

Cosine Function
The cosine function, often written as \( \cos(x) \), is a fundamental part of trigonometry. It connects the angle of a right triangle with the ratio of the adjacent side over the hypotenuse.
  • Angle and Ratio: In a unit circle, the cosine of an angle represents the x-coordinate of the point on the circle at that angle.
  • Range and Periodicity: Cosine values range from -1 to 1, showing its periodic nature, repeating every \(2\pi\).
  • Even Function: Cosine is an even function, which means \(\cos(-x) = \cos(x)\).
Understanding the cosine function helps in solving various trigonometric problems, especially when working with identities and equations.
Double Angle Formula
The Double Angle Formula is a useful trigonometric identity that relates the cosine of a double angle to the cosine of a single angle. In its common form for cosine, it is expressed as:
\[ \cos(2x) = 2 \cos^2 x - 1 \]
This formula allows you to simplify expressions involving \(\cos(2x)\) in terms of \(\cos(x)\), which is essential in problems like the one we are dealing with for \(\cos(4x)\).
  • By using the formula, you replace a more complex expression with a simpler one.
  • This formula can be applied repeatedly, as seen in the steps of rewriting \(\cos 4x\) in terms of \(\cos(x)\).
  • Remember, a double angle leads to multiplying the angle by two, hence the name "double angle."
Trigonometric Simplification
Trigonometric simplification involves reducing complex trigonometric expressions into simpler or more recognizable forms. This helps by making calculations easier and clearer.
  • Applying Identities: Use identities such as the double angle or Pythagorean identities to transform the expressions.
  • Stepwise Process: In the exercise provided, each step was important to transform \(\cos 4x\) into \(8 \cos^4 x - 8 \cos^2 x + 1\).
  • Rewriting Sub-expressions: Remember, expressing all terms in terms of a single variable like \(\cos(x)\) simplifies the handling of the expression.
Simplification is key because it allows deeper understanding and often reveals hidden relationships within trigonometric expressions.

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Most popular questions from this chapter

Use the Quadratic Formula to solve the equation in the interval \(0,2 \pi\). Then use a graphing utility to approximate the angle \(x\). $$4 \cos ^{2} x-4 \cos x-1=0$$

Find the exact value of the trigonometric expression given that \(\sin u=-\frac{7}{25}\) and \(\cos v=-\frac{4}{5} .\) (Both \(u\) and \(v\) are in Quadrant III.$$\sec (v-u)$$

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Use a graphing utility to graph \(y_{1}\) and \(y_{2}\) in the same viewing Window. Use the graphs to determine whether \(y_{1}=y_{2}\).Explain your reasoning.$$y_{1}=\cos (x+2), \quad y_{2}=\cos x+\cos 2$$.
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