Question 4.6: The Cartesian coordinates of P are (4, 7); those of Q are (−......
The Cartesian coordinates of P are (4, 7); those of Q are (−5, 6). Calculate the polar coordinates of P and Q.
Figure 4.13 illustrates the situation for P.
Then
r=\sqrt{4^2+7^2}=\sqrt{65}=8.0623
Note from Figure 4.13 that P is in the first quadrant, that is θ lies between 0° and 90°. Now
\tan ^{-1}\left(\frac{y}{x}\right)=\tan ^{-1}\left(\frac{7}{4}\right)
From a calculator, \tan ^{-1}\left(\frac{7}{4}\right)=60.26^{\circ}. lies between 0° and 90° then clearly 60.26° is the required value.
The polar coordinates of P are r = 8.0623, θ = 60.26°.
Figure 4.14 illustrates the situation for Q.
We have
r=\sqrt{(-5)^2+6^2}=\sqrt{61}=7.8102
From Figure 4.14 we see that lies between 90° and 180°. Now
\tan ^{-1}\left(\frac{y}{x}\right)=\tan ^{-1}\left(\frac{6}{-5}\right)=\tan ^{-1}(-1.2)
A calculator returns the value of −50.19° which is clearly not the required value. Recall that \tan \theta is periodic with period 180°. Hence the required angle is
180^{\circ}+\left(-50.19^{\circ}\right)=129.81^{\circ}
The polar coordinates of Q are r = 7.8102, θ = 129.81°.
