Question

How do you find the exact values of θ for
cscθ= 2 √3 / 3

Answer 1

replace cscθ with 1/sinθ
so you have
\[ \frac{1}{\sin \theta}= \frac{2 \sqrt{3}}{3} \]

now flip both sides.
what do you get ?

Answer 2

sinθ= 1/ 2 √3 / 3

Answer 3

you flip looks confusing...
don't you mean
\[ \sin θ = \frac{3}{2\sqrt{3}} \]

now divide sqr(3) into 3. what do you get ?

Answer 4

can you explain the flip...

Answer 5

the denominator becomes the numerator and the numerator becomes the denominator

Answer 6

if you start with
\[ \frac{a}{b} \]
to flip, you make the top the bottom and vice versa
\[\frac{b}{a} \]

Answer 7

so in you problem the 2 sqr3 becomes the bottom
and 3 becomes the top

Answer 8

now the next step is simplify 3/sqr(3)

Answer 9

Got it. :)

Answer 10

you should remember that
sqr3 * sqr3 = 3
if you divide both sides by sqr3 you get
sqr3 = 3/sqr3

use that to simplify
\[ \sin \theta= \frac{3}{2\sqrt{3}} \]

Answer 11

okay so cscθ = 2√3/3 would be equal to sinθ= 3/ 2√3

Answer 12

yes, but you should simplify to
\[ \sin\theta= \frac{\sqrt{3}}{2} \]
that should look familiar.
see http://www.easycalculation.com/trigonometry/trigonometry-tables.php

Answer 13

can you show me how you got from 3/ 2√3 to √3/2.

Answer 14

I did. the idea is
\[ 3=\sqrt{3} \cdot \sqrt{3}\]
that is the definition of square root

if we divide both sides by sqr3 we get
\[ \frac{3}{\sqrt{3}}= \sqrt{3} \]
that means you can replace \( \frac{3}{\sqrt{3}}\) with \(\sqrt{3} \)

Answer 15

another way to simplify is "rationalize" the fraction by multiplying top and bottom by sqr(3)
\[ \frac{3}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}= \frac{3\sqrt{3}}{2\cdot 3}
= \frac{\cancel{3}\sqrt{3}}{2\cdot \cancel{3}}= \frac{\sqrt{3}}{2}
\]

Answer 16

That works better for me. ;) Okay so then I know sin(x)= √3/2 which has an exact value or 60 degrees.
If the given domain is -270 degrees<x<180 degrees I can find the other solutions on the unit circle correct?

Answer 17

another one would be 120 degrees because sin is positive in that quadrant as well.

Answer 18

is it -270º ? if so, that is the same as 90º
and you want the angle in the 2nd quadrant (90 to 180)

the reference angle is 60, and the angle you want is 180-60=120º

Answer 19

Okay I got the 120 degrees. so what would the exact value be for the - 270 degrees? Wouldn't it be a negative value... so -90 degrees?

Answer 20

minus 270 means start at the x-axis and rotate clockwise 270 degrees, you end up at +90

or
-270+360 (you can always add 360º)

-90 is the same as 270º
(numerically: -90+360)

Answer 21

but how would you state that as an exact value

Answer 22

I don't understand the question.

I thought it was:
if we restrict the domain to -270º to 180º what theta gives csc(theta)=2 √3 / 3

Answer 23

sorry. This isn't explained very well in the textbook.
One of the values is 60 degrees and the other is 120 degrees. I understand that completely, but what about the negative value?

Answer 24

if they really mean -270º to +180º then that includes
-240º
+60º
+120º

Answer 25

-240 because there's the negative value of √3/2 on the unit circle. It's all coming together. I really appreciate your time.

Answer 26

Now to go one step further and determine the exact values of cos(x) and tan(x) I would use their ratios, right?

Answer 27

**-240 because there's the negative value of √3/2 on the unit circle.***

I would not say it like that... it sounds wrong
you only want angles A such that sin(A)= +√3/2