Answered

Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.

Arrange the steps in the correct order to express [tex]\cos 3x[/tex] in terms of [tex]\cos x[/tex].

[tex]\[
\begin{array}{|c|}
\hline
\cos(2x + x) \\
\cos(2x)\cos(x) - \sin(2x)\sin(x) \\
[1 - 2\sin^2(x)]\cos(x) - [2\sin(x)\cos(x)]\sin(x) \\
\cos(x) - 4\sin^2(x)\cos(x) \\
\cos(x)[1 - 4\sin^2(x)] \\
\cos(x)[1 - 4(1 - \cos^2(x))] \\
\cos(x)[-3 + 4\cos^2(x)] \\
4\cos^3(x) - 3\cos(x) \\
\hline
\end{array}
\][/tex]

Sagot :

GinnyAnswer
To express [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex], we need to arrange the given steps in the correct logical order. Here is the detailed, step-by-step process:

1. Start with the angle sum formula for cosine:
[tex]\[ \cos(3x) = \cos(2x + x) \][/tex]

2. Apply the angle addition formula for cosine:
[tex]\[ \cos(3x) = \cos(2x) \cos(x) - \sin(2x) \sin(x) \][/tex]

3. Express [tex]\(\cos(2x)\)[/tex] and [tex]\(\sin(2x)\)[/tex] in terms of [tex]\(\cos(x)\)[/tex] and [tex]\(\sin(x)\)[/tex]:
[tex]\[ \cos(2x) = 1 - 2\sin^2(x) \quad \text{and} \quad \sin(2x) = 2 \sin(x) \cos(x) \][/tex]
Substitute these into the formula:
[tex]\[ \cos(3x) = (1 - 2\sin^2(x)) \cos(x) - (2 \sin(x) \cos(x)) \sin(x) \][/tex]

4. Simplify the expression:
[tex]\[ \cos(3x) = [1 - 2\sin^2(x)] \cos(x) - [2 \sin(x) \cos(x)] \sin(x) \][/tex]
[tex]\[ \cos(3x) = \cos(x) - 2\sin^2(x) \cos(x) - 2\sin(x) \cos(x) \sin(x) \][/tex]
[tex]\[ \cos(3x) = \cos(x) - 4\sin^2(x) \cos(x) \][/tex]

5. Rewrite [tex]\(\sin^2(x)\)[/tex] as [tex]\(1 - \cos^2(x)\)[/tex]:
[tex]\[ \cos(3x) = \cos(x) - 4(1 - \cos^2(x)) \cos(x) \][/tex]
[tex]\[ \cos(3x) = \cos(x) - 4[\sin^2(x)] \cos(x) \][/tex]
[tex]\[ \cos(3x) = \cos(x)\left\{1 - 4[1 - \cos^2(x)]\right\} \][/tex]

6. Simplify further:
[tex]\[ \cos(3x) = \cos(x)\left[1 - 4 + 4\cos^2(x)\right] \][/tex]
[tex]\[ \cos(3x) = \cos(x)\left[-3 + 4\cos^2(x)\right] \][/tex]

7. Final form:
[tex]\[ \cos(3x) = 4\cos^3(x) - 3\cos(x) \][/tex]

Therefore, the correct order of steps to express [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex] is:

1. [tex]\(\cos(2x + x)\)[/tex]
2. [tex]\(\cos(2x) \cos(x) - \sin(2x) \sin(x)\)[/tex]
3. [tex]\([1-2 \sin^2(x)] \cos(x) - [2 \sin(x) \cos(x)] \sin(x)\)[/tex]
4. [tex]\(\cos(x) - 4 \sin^2(x) \cos(x)\)[/tex]
5. [tex]\(\cos(x)[1 - 4 \sin^2(x)]\)[/tex]
6. [tex]\(\cos(x)\{1 - 4 [1 - \cos^2(x)]\}\)[/tex]
7. [tex]\(\cos(x)[-3 + 4 \cos^2(x)]\)[/tex]
8. [tex]\(4 \cos^3(x) - 3 \cos(x)\)[/tex]

This completes the detailed, step-by-step solution for expressing [tex]\(\cos 3x\)[/tex] in terms of [tex]\(\cos x\)[/tex].
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.