Lesson 10 Complex Polar Planes Solidify Understanding

Ready

Transform point $A$ as indicated below.

1.

Apply the rule $(x, y) \to (x - 2, y - 5)$ to point $A$ . Label as $A^{'}$
Apply the rule $(x, y) \to (x - 1, y + 3)$ to point $A^{'}$ . Label as $A^{''}$ .
Apply the rule $(x, y) \to (- 2 x, y)$ to point $A^{''}$ . Label as $A^{'''}$ .

Transform the given graph as indicated below.

2.

Apply the rule $(x, y) \to (x - 3, y)$ to $f (x)$ . Label as $f^{'} (x)$ .
Apply the rule $(x, y) \to (x, y - 5)$ to $f (x)$ . Label as $f^{''} (x)$ .
Apply the rule $(x, y) \to (x, \frac{1}{2} y)$ to $f (x)$ . Label as $f^{'''} (x) .$

Set

Coordinate Conversion:

The polar coordinates $(r, θ)$ are related to the rectangular coordinates $(x, y)$ as follows:

$x = r \cos θ$
$y = r \sin θ$
$\tan θ = \frac{y}{x}$
$r^{2} = x^{2} + y^{2}$

Convert the points from polar to rectangular coordinates.

3.

$(4, \frac{π}{2})$

4.

$(\sqrt{3}, \frac{5 π}{6})$

5.

$(2, π)$

6.

$(5 \sqrt{2}, \frac{7 π}{4})$

Convert the points from rectangular to polar coordinates.

7.

$(- 3, 3)$

8.

$(- 6, 0)$

9.

$(1, \sqrt{3})$

10.

$(- 4 \sqrt{3}, - 4)$

Consider the complex number $a + b i$ . The angle $θ$ is the angle measure from the positive real axis to the line segment connecting the origin and the point $(a, b)$ . $a = r \cos θ$ and $b = r \sin θ$ , where $r = \sqrt{a^{2} + b^{2}}$ .

By replacing $a$ and $b$ , you have $a + b i = (r \cos θ) + (r \sin θ) i$ . Factor out the $r$ to obtain the trigonometric form of a complex number.

If $z = a + b i$ , then the trigonometric form is $z = r (\cos θ + i \sin θ)$ .

Write the complex numbers in trigonometric form $z = r (\cos θ + i \sin θ)$ .

11.

$- 3 - i$

12.

$3 - 3 i$

13.

$- 7 + 4 i$

14.

$\sqrt{3} + i$

Write the complex number in standard form $a + b i$ .

15.

$3 (\cos 120 ° + \sin 120 °)$

16.

$5 (\cos 135 ° + i \sin 135 °)$

17.

$\sqrt{8} (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})$

18.

$8 (\cos \frac{π}{2} + i \sin \frac{π}{2})$

Go

Use the definition of $\log x$ to find the value of $x$ . Recall that $\log x$ has a base of $10$ . (NO CALCULATORS)

19.

$\log x = 3$

20.

$\log x = - 4$

21.

$\log x = 1$

22.

$\log x = 0$

23.

$\log x = \frac{1}{2}$

24.

$\log 10^{7} = x$

25.

$\log 0.001 = x$

26.

$\log 1,000,000 = x$

27.

$\log 10^{e} = x$