Question

Find all values of theta where 0 degrees < 0 < 360 degrees.

a. csc(theta) = - √2
b. sin(theta) = -√3/2
c. sec(theta) = -1

Answer 1

how far did you get?

Answer 2

Nothing.. Completely lost and I need to finish this asap.

Answer 3

csc(theta) = 1/sin(theta)

Answer 4

csc(theta) = - √2

1/sin(theta) = - √2

sin(theta) = - 1/√2

sin(theta) = -(√2 )/2

now use the unit circle to determine theta

Answer 5

I'll let you take over. Tell me what you get

Answer 6

Um.. 4pi/3?

Answer 7

close but no

Answer 8

But -1/2 , -√2, 2 isn't stated on the unit circle. I'm unsure how to find the measurement.

Answer 9

try looking for \(\large -\frac{\sqrt{2}}{2}\)

Answer 10

That's in quadrants 3 & 4.

Answer 11

yes because sine is negative here

Answer 12

which angles correspond to a point with a y coordinate of \(\large -\frac{\sqrt{2}}{2}\)

Answer 13

225 , 315

Answer 14

very good, you got them both

Answer 15

so that is the solution to part A

Answer 16

Oh okay that makes much more sense!

Answer 17

you are doing the same for each part

note: sec(theta) = 1/cos(theta)

Answer 18

that's great

Answer 19

Can you set the problems up again? Still confusing in that area.

Answer 20

which one

Answer 21

B

Answer 22

in part B you need to look at the unit circle for points that have y coordinates of \(\large -\frac{\sqrt{3}}{2}\)

Answer 23

then report the angles (in degrees) for those corresponding points

Answer 24

remember, sine deals with the y coordinate on the unit circle

Answer 25

240 , 300?

Answer 26

both are perfect

Answer 27

Awesome, what about C?

Answer 28

sec(theta) = -1

1/cos(theta) = -1

cos(theta) = -1

now use the unit circle...BUT this time you're looking at the x coordinate (not y coordinate)

Answer 29

180, ?

Answer 30

and that's it, just 180

Answer 31

Awesome, thank you so much! Do you mind helping me out with coterminals?

Answer 32

sure I can do a few more

Answer 33

Find two angles, theta, coterminal with -25pi/8 where -2pi<theta<2pi.

Answer 34

use a calculator to evaluate -25pi/8

what do you get

Answer 35

-9.817

Answer 36

notice how -2pi = -6.28 roughly

Answer 37

and 2pi = 6.28 roughly

Answer 38

so clearly -25pi/8 is not in the interval -2pi<theta<2pi

Answer 39

Okay

Answer 40

to find any coterminal angles to a given angle (in radians) we just add/subtract 2pi

Answer 41

in this case, we add 2pi

Answer 42

-25pi/8 + 2pi = ??

Answer 43

Um.. -23pi/8?

Answer 44

no, 2pi = 4pi/2

Answer 45

Oh one sec

Answer 46

sorry I meant to say 2pi = 16pi/8

Answer 47

-25pi/8 + 2pi

-25pi/8 + 16pi/8 = ??

Answer 48

-9pi/8?

Answer 49

now evaluate that with a calculator

Answer 50

-3.53

Answer 51

is that in the interval

-6.28 < theta < 6.28

??

Answer 52

Yes it is :)

Answer 53

What about the other one?

Answer 54

so that's one coterminal angle that's in the right interval we want

add 2pi to -9pi/8 to get the next coterminal angle

Answer 55

7pi/8?

Answer 56

very good, you got the hang of this

Answer 57

2.74

Answer 58

so that's definitely in range too

Answer 59

the two coterminal angles are

-9pi/8, 7pi/8

Answer 60

Great thanks! I have 2 more different problems to solve rather quickly because I can figure the rest off the bases of these. Do you mind? You're a great help.

Answer 61

sure go for it

Answer 62

Find the measure of the reference angle:
a. theta = 18pi/7

Find the radian measure of the angle:
theta = -270

degree measure of :
theta=8pi/12

Sorry 3 actually.

Answer 63

evaluate 18pi/7 with a calculator

Answer 64

8.07

Answer 65

so that's definitely out of range of 0 < theta < 2pi

Answer 66

so we need to find the coterminal angle

subtract 2pi from 18pi/7 to get

18pi/7 - 2pi = ???

Answer 67

Hmm the denominators are different..

Answer 68

2pi = 14pi/7

Answer 69

multiply top and bottom by 7

Answer 70

Oh okay. So 4pi/7.

Answer 71

4pi/7 = 1.795

so 4pi/7 is definitely in the interval 0 < theta < 2pi

Answer 72

Oh wait. This is to find the reference angle.

Answer 73

the question is: which quadrant is 4pi/7 in?

well we could do a bunch of computations of pi/2, pi, 3pi/2 and compare, or we can convert 4pi/7 to degrees

Answer 74

I would convert it over to degrees

so you multiply this by 180/pi to get

(4pi/7)*(180/pi) = (4pi*180)/(7*pi) = 720/7 = 102.857

now what quadrant is the angle 102.857 degrees in?

Answer 75

2?

Answer 76

good

Answer 77

so we use this rule

Reference Angle = pi - (Given Angle)

Answer 78

Reference Angle = pi - (Given Angle)

Reference Angle = pi - 4pi/7

Reference Angle = 7pi/7 - 4pi/7

Reference Angle = 3pi/7

Answer 79

Makes sense! Awesome!

Answer 80

ok that's great

Answer 81

Now for question 2

Answer 82

Find the radian measure of the angle: theta = -270

Answer 83

multiply this by pi/180 and reduce

Answer 84

-540?

Answer 85

no

-270*(pi/180) = ??

Answer 86

-270pi/-1.5 ? Er.

Answer 87

-270*(pi/180) = -270pi/180 which reduces to what?

Answer 88

-4.7

Answer 89

leave it as a fraction though

Answer 90

it might help to just focus on reducing -270/180

Answer 91

-9pi/4?

Answer 92

close, it should be -3pi/2

Answer 93

now add 2pi to this

Answer 94

pi/2?

Answer 95

very good

Answer 96

:) Question 3 and I'll leave ya alone, :)

Answer 97

degree measure of : theta=8pi/12

Answer 98

multiply this by 180/pi and reduce