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Chapter 11: Problem 17

Express the following polar coordinates in Cartesian coordinates. $$\left(1,-\frac{\pi}{3}\right)$$

Short Answer

Expert verified
Answer: The Cartesian coordinates for the given polar coordinates are (1/2, -鈭�3/2).

Step by step solution

01

Identify the given polar coordinates

We are given the polar coordinates \((r, \theta) = \left(1, -\frac{\pi}{3}\right)\).
02

Convert polar to Cartesian coordinates

Use the conversion equations with the given polar coordinates to find the Cartesian coordinates. For the x-coordinate, we have: $$x = r \cos(\theta) = 1 \cos\left(-\frac{\pi}{3}\right)$$ For the y-coordinate, we have: $$y = r \sin(\theta) = 1 \sin\left(-\frac{\pi}{3}\right)$$
03

Evaluate the trigonometric functions

Calculate the values of \(\cos\left(-\frac{\pi}{3}\right)\) and \(\sin\left(-\frac{\pi}{3}\right)\). Recall that negative angles in trigonometric functions mean rotation in the clockwise direction. In this case, we can rewrite the angles as follows: $$\cos\left(-\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{3}\right)$$ $$\sin\left(-\frac{\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right)$$ Now, we know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\) and \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), so we have: $$\cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}$$ $$\sin\left(-\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$$
04

Calculate the Cartesian coordinates

Plug in these values into the x and y equations to obtain the Cartesian coordinates. $$x = 1 \cdot \frac{1}{2} = \frac{1}{2}$$ $$y = 1 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{\sqrt{3}}{2}$$ The Cartesian coordinates for the given polar coordinates are \(\left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right)\).

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