Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter6: Rates Of Change
Section6.1: Velocity
Problem 12SBE
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![### Mathematical Proof of a Trigonometric Identity
#### Given:
\[ \tan u = \frac{\partial u / \partial y}{\partial u / \partial x} \]
#### To Prove:
\[ \left(\frac{\partial r^{2}}{\partial x}\right)^{2} + \left(\frac{\partial r^{2}}{\partial y}\right)^{2} = r^{2} \left(\frac{\partial r^{2}}{\partial y}\right)\]
In this proof, we start with the trigonometric function tangent (tan), which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The given equation involves partial derivatives of a function `u` with respect to the variables `x` and `y`.
**Explanation of Work:**
1. **Original Equation:**
\[ \tan u = \frac{\frac{\partial u}{\partial y}}{\frac{\partial u}{\partial x}} \]
2. **Objective:**
\[ \left( \frac{\partial r^{2}}{\partial x} \right)^{2} + \left( \frac{\partial r^{2}}{\partial y} \right)^{2} = r^{2} \left( \frac{\partial r^{2}}{\partial y} \right) \]
*Note: Explanation assumes partial derivatives and uses squared terms to potentially simplify and manipulate to the desired equation.*
In summary, by analyzing the given function and using differentiation techniques, we aim to validate the given identity through a series of mathematical manipulations. This involves understanding the relationship between the trigonometric function and partial derivatives in a geometrical context.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb84913bb-ff7f-4744-9adc-3017f6c2bef1%2F065c1980-17fd-4dfa-ba2b-5ee03ecee6b5%2F552na4.jpeg&w=3840&q=75)
Transcribed Image Text:### Mathematical Proof of a Trigonometric Identity
#### Given:
\[ \tan u = \frac{\partial u / \partial y}{\partial u / \partial x} \]
#### To Prove:
\[ \left(\frac{\partial r^{2}}{\partial x}\right)^{2} + \left(\frac{\partial r^{2}}{\partial y}\right)^{2} = r^{2} \left(\frac{\partial r^{2}}{\partial y}\right)\]
In this proof, we start with the trigonometric function tangent (tan), which is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. The given equation involves partial derivatives of a function `u` with respect to the variables `x` and `y`.
**Explanation of Work:**
1. **Original Equation:**
\[ \tan u = \frac{\frac{\partial u}{\partial y}}{\frac{\partial u}{\partial x}} \]
2. **Objective:**
\[ \left( \frac{\partial r^{2}}{\partial x} \right)^{2} + \left( \frac{\partial r^{2}}{\partial y} \right)^{2} = r^{2} \left( \frac{\partial r^{2}}{\partial y} \right) \]
*Note: Explanation assumes partial derivatives and uses squared terms to potentially simplify and manipulate to the desired equation.*
In summary, by analyzing the given function and using differentiation techniques, we aim to validate the given identity through a series of mathematical manipulations. This involves understanding the relationship between the trigonometric function and partial derivatives in a geometrical context.
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