Chapter 9: Q 58. (page 757)

Complete the proof of Theorem 9.14 by verifying that
${(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2}$ is equal to ${(f' (θ))}^{2} + {(f (θ))}^{2}$

Short Answer

Expert verified

It is proved that ${(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2} = {(f' (θ))}^{2} + {(f (θ))}^{2}$

Step by step solution

Step 1. Given information

Left side of the equation is ${(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2}$

Step 2. Explanation

$\begin{array}{rcl} {(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2} & = & {(f' (θ))}^{2} (\cos^{2} θ + \sin^{2} θ) - 2 f' (θ) f (θ) \cos θ \sin θ + {(f (θ))}^{2} (\sin^{2} θ + \cos^{2} θ) \\ + 2 f' (θ) f (θ) \cos θ \sin θ \\ = & {(f' (θ))}^{2} + {(f (θ))}^{2} \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Complete the proof of Theorem 9.14 by verifying that
${(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2}$ is equal to ${(f' (θ))}^{2} + {(f (θ))}^{2}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Probability and Statistics

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Complete the proof of Theorem 9.14 by verifying thatf'θcosθ-fθsinθ2+f'θsinθ+fθcosθ2is equal tof'θ2+fθ2

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Statistics

Probability and Statistics

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Complete the proof of Theorem 9.14 by verifying that
${(f' (θ) \cos θ - f (θ) \sin θ)}^{2} + {(f' (θ) \sin θ + f (θ) \cos θ)}^{2}$ is equal to ${(f' (θ))}^{2} + {(f (θ))}^{2}$