Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 9 - Section 9.4 - Rotation of Axes - Exercise Set - Page 1010: 13


$x= \frac{1}{5}(3x'-4y')$ and $y= \frac{1}{5}(4x'+3y')$

Work Step by Step

Step 1. From the given equation, we have $A=10, B=24, C=17$. Thus we have $cot2\theta=\frac{A-C}{B}=\frac{10-17}{24}=-\frac{7}{24}$ Thus $2\theta$ is in Quadrant II and $cos2\theta=-\frac{7}{\sqrt {7^2+24^2}}=-\frac{7}{25}$ (form a right triangle with sides $7,24,25$ here). Step 2. We can find values of $sin\theta$ and $\cos\theta$ as $sin\theta=\sqrt {\frac{1-cos2\theta}{2}}=\sqrt {\frac{1-(-\frac{7}{25})}{2}}=\frac{4}{5}$ and $cos\theta=\sqrt {1-sin^2\theta}=\frac{3}{5}$ Step 3. Using the transformation formula $x=x'cos\theta-y'sin\theta$ and $y=x'sin\theta+y'cos\theta$, we have $x=x'(\frac{3}{5})-y'(\frac{4}{5})=\frac{1}{5}(3x'-4y')$ $y=x'(\frac{4}{5})+y'(\frac{3}{5})=\frac{1}{5}(4x'+3y')$
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