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Chapter 6: Problem 116

Write the trigonometric expression as an algebraic expression. $$\cos (2 \arccos x)$$

Short Answer

Expert verified
The trigonometric expression \( \cos (2 \arccos x) \) can be written as an algebraic expression as \( 2x^2 - 1 \).

Step by step solution

01

Identify the double-angle identity

The relevant identity is \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). Here, \( \theta \) is \( \arccos x \).
02

Replace sin in the formula

Since \( \arccos x\) is the angle in our formula, we need to express \( \sin(\arccos x) \) using an algebraic expression. We know that \( \sin^2(\theta) + \cos^2(\theta) = 1 \), so \( \sin(\arccos x) \) can be expressed as \( \sqrt{1 - x^2} \). Therefore, \( \sin^2(\arccos x) = 1 - x^2 \).
03

Substitute into the original formula

Substitute the value of \( \sin^2(\arccos x) \) into the formula, we will get \( \cos (2 \arccos x) = 1 - 2(1 - x^2) \). Simplifying this gives: \( 2x^2 - 1 \).

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Key Concepts

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

Double-Angle Identities
Double-angle identities are crucial in transforming and simplifying trigonometric expressions. They are relationships that involve trigonometric functions of double angles, such as \( \cos(2\theta) \) or \( \sin(2\theta) \).

In this exercise, we use the identity \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). This helps express \( \cos(2\theta) \) in terms of another trigonometric function, \( \sin \). These identities are especially useful in solving trigonometric equations and converting them to algebraic forms.
  • Identities reduce complexity by simplifying expressions.
  • They allow replacement of higher powers and angles with simpler terms.
Learning these identities improves problem-solving skills in mathematics.
Algebraic Expressions
Algebraic expressions involve numbers and variables combined using operations like addition, subtraction, and multiplication. They are often used to represent regular arithmetic operations algebraically.

In this solution, we transform the trigonometric function \( \cos(2 \arccos x) \) into an algebraic expression. This involves substituting \( \sin^2(\arccos x) \) with a square root expression derived from the Pythagorean identity.
  • The expression \( \sqrt{1 - x^2} \) comes from \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
  • Replacing trigonometric components helps simplify the final expression to \( 2x^2 - 1 \).
Through these conversions, complex trigonometric equations become more manageable.
Inverse Trigonometric Functions
Inverse trigonometric functions allow determination of angles given numerical values of trigonometric functions. For example, \( \arccos x \) is the angle whose cosine is \( x \).

In our problem, the process starts with expressing \( \arccos x \) and converting it through inverse functions, which connect angles to their trigonometric values. Understanding \( \arccos x \) helps transform \( \cos(2 \arccos x) \) by using the identity formula.
  • These functions allow backtracking from a ratio to an angle.
  • This conversion plays a critical role in achieving the final algebraic form.
Knowing how to manipulate inverse trigonometric functions widens the scope to solve more complex trigonometric problems and improves mathematical fluency.

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Most popular questions from this chapter

Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. $$\frac{\pi}{12}$$

Find the exact values of \(\sin (u / 2), \cos (u / 2),\) and \(\tan (u / 2)\) using the half-angle formulas. $$\csc u=-\frac{5}{3}, \quad \pi

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin (\alpha+\beta)-\sin (\alpha-\beta)$$

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Perform the addition or subtraction and simplify. $$\frac{2 x}{x^{2}-4}+\frac{5}{x+4}$$
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