Lesson 8 The Amazing Inverse Trig Function Race Solidify Understanding

Ready

Plot each number in the rectangular complex plane.

1.

$7 + 5 i$

2.

$- 4 - i$

3.

$- 6 i + 2$

4.

$8 - 6 i$

5.

Use the Pythagorean Theorem to find the modulus of each number in problems 1–4. (Recall that the modulus is the distance between the point $(0, 0)$ and the point representing the number.)

a.

modulus of $7 + 5 i$

b.

modulus of $- 4 - i$

c.

modulus of $- 6 i + 2$

d.

modulus of $8 - 6 i$

6.

Multiply each complex number in problems 1–4 by its conjugate.

a.

Multiply $7 + 5 i$ by its conjugate.

b.

Multiply $- 4 - i$ by its conjugate.

c.

Multiply $- 6 i + 2$ by its conjugate.

d.

Multiply $8 - 6 i$ by its conjugate.

e.

Compare your answers to the ones you got in problem 5. What do you notice?

Set

Use the given information to find the missing angle $(0 \leq θ \leq 2 π)$ .

Round answers to the thousandths place ( $3$ decimal places).

7.

$\cos θ = 0.9848$ ; $\sin θ > 0$

8.

$\cos θ = 0.9848$ ; $\sin θ < 0$

9.

Explain why the answers to problems 7 and 8 are different even though the calculator gives you the same answer for both problems. (Include information about the meaning of the inverse trig function and the quadrant in which the terminal ray resides.)

10.

$\sin θ = - 0.1908$ ; $\tan θ < 0$

11.

$\sin θ = - 0.1908$ ; $\tan θ > 0$

12.

Explain why the answers to problems 10 and 11 are different even though the calculator gives you the same answer for both problems. (Include information about the meaning of the inverse trig function and the quadrant in which the terminal ray resides.)

13.

$\tan θ = - 0.4663$ ; $\sin θ > 0$

14.

$\tan θ = - 0.4663$ ; $\cos θ > 0$

15.

Explain why the answers to problems 13 and 14 are different even though the calculator gives you the same answer for both problems. (Include information about the meaning of the inverse trig function and the quadrant in which the terminal ray resides.)

16.

$\sin θ = - 1$

17.

Explain why problem 16 needed only one clue to determine a unique value for $θ$ , while the other problems required at least two clues.

Fill in the indicated information for each of the inverse functions.

18.

a.

Graph $y = \sin^{- 1} x$ .

b.

domain:

c.

range:

19.

a.

Graph $y = \cos^{- 1} x$ .

b.

domain:

c.

range:

20.

a.

Graph $y = \tan^{- 1} x$ .

b.

domain:

c.

range:

Go

Plot the given point $(r, θ)$ in the polar coordinate system.

21.

$(r, θ) = (5, \frac{2 π}{3})$

22.

$(r, θ) = (3, \frac{7 π}{6})$

23.

$(r, θ) = (8, \frac{5 π}{3})$

24.

$(r, θ) = (9, \frac{π}{3})$

25.

$(r, θ) = (7, \frac{5 π}{6})$

26.

$(r, θ) = (- 6, \frac{π}{3})$