Lesson 9 More Hidden Identities Practice Understanding

Ready

In the diagram, $3$ complex numbers have been graphed as vectors. Rewrite each complex number as a point in the form $(a, b)$ .

1.

$- 3 + 4 i$ graphs as $(\underset{―}{}, \underset{―}{})$

2.

$5 + 2 i$ graphs as $(\underset{―}{}, \underset{―}{})$

3.

$1 - 6 i$ graphs as $(\underset{―}{}, \underset{―}{})$

4.

Graph the complex numbers as vectors on the diagram.

$- 5 - 3 i$
$2 + 4 i$
$- 6 + i$
$2 - i$

5.

The magnitude of a complex is its modulus. It is symbolized by the notation $| a + b i |$ where $| a + b i | = \sqrt{a^{2} + b^{2}}$ .

Find the modulus of each of the complex numbers.

a.

$- 3 + 4 i$

b.

$5 + 2 i$

c.

$1 - 6 i$

d.

$- 5 - 3 i$

e.

$2 + 4 i$

f.

$- 6 + i$

g.

$2 - i$

Set

Solve the following trigonometric equations. Write your answer(s) in the form $θ \pm 2 π k$ , where $2 π$ represents the interval between successive solutions and $k$ is any integer. (Note: sometimes the next solution can be described as just $\pm π k$ or some other interval, instead of $2 π k$ .)

6.

$5 \cos x + \sqrt{2} = \sqrt{2}$

7.

$2 \sin x + \sqrt{2} = 0$

8.

$\tan x + 2 \sqrt{3} = 3 \sqrt{3}$

9.

$\sin^{2} x = 3 \cos^{2} x$

10.

$2 - \sin^{2} x = 1 + \cos x$

11.

$2 \sin^{2} x + 3 \sin x + 1 = 0$

12.

$2 \sin^{2} (2 x) = 1$

13.

$2 \cos^{2} x - \cos x = 1$

14.

$2 \cos^{2} x - 5 \cos x + 2 = 0$

Go

Use the graph to find all of the values for $x$ when $y = 0$ , for the given equation. Write your answer(s) in the form $+ n k$ , where $n k$ represents the interval between successive solutions.

15.

$y = \sin (\frac{π x}{2}) + 1$

16.

Use the unit circle to explain the solutions you found in problem 15.

17.

Use the graph to approximate the points of intersection of the graphs of $y_{1}$ and $y_{2}$ .

$y_{1} = 2 \sin x + 1$

$y_{2} = \frac{1}{3} x + 2$

18.

The scale on the $x$ -axis in the graph of problem 18 is $1$ . The scale in the graph of problem 20 is $\frac{π}{2}$ . Yet the units on both axes is radians.

a.

Label the graph with the approximate location of $\frac{π}{2}$ and $π$ .

b.

Label the graph with the approximate location of $1$ , $2$ , $3$ , and $4$ .