Question

An equation is given. (a) Find all solutions. (b) Find the solutions in the interval [0,2pi)

csc 3theta = 5sin 3theta

Answer 1

This is what I got so far.

csc 3x = 5sin3x

1/sin3x = 5sin3x/1
1 = (sin3x)(5sin3x)
1 = 5sin^(2)3x
+/- (1/√5) = sin3x

sin3x= 1/√5
sinx = 3/ √5
BUT I got an error!

What am I doing wrong?

Answer 2

the last step is incorrect as the 3x is inside the sine function so you cannot just divide by 3, you would have to do sin inverse on both sides first.

Answer 3

So the correct step would be: 3x=sin^(-1)(1/sqrt(5))

Answer 4

@amyhashimoto77 have you solved it yet?

Answer 5

You have done a great job except your last line.

sin3x = + or - \[\frac{ 1 }{ \sqrt{5} }\]

now let's define "alpha" = sin^(-1)(\[\frac{ 1 }{ \sqrt{5} }\]) (you can take any letter)

3x = pi*n +/- alpha

x = pi*n/3 +/alpha/3

Answer 6

Me?

Answer 7

It didn't show how I typed

Answer 8

@lochana did I make a mistake?

Answer 9

No. You are also correct. But conventionally ,we must add pi*n in front of your solution.

Answer 10

Oh yes I remember that. Sorry I forgot!

Answer 11

@amyhashimoto77 have you worked out how to solve it yet?

Answer 12

I'm leaving, notify me if you have any questions.

Answer 13

Actually I just did big mistake here. apologies for that first of all.
please read this first, It will show you how to write general solutions for sin. I did this at my school long ago. So I forgot it. sorry
http://www.math-only-math.com/sin-theta-equals-sin-alpha.html

Answer 14

so the correct solution is

x = pi*n/3 + (-1)^n (alpha/3) n is an integer.

Answer 15

The answer for your second question is
\[x = \sin^{-1} {1/\sqrt{5}}\]
\[x = -\sin^{-1} {1/\sqrt{5}}\]

these are the only solutions exit on the graph.which is in the interval [0,2pi)

Answer 16

@amyhashimoto77 are you still having problems?

Answer 17

@lochana @seascorpion1
Thank you so much for your help.

So this is what i got for the last part .

sin3x = 1/√5
sin^(-1)(sin3x) = sin^(-1)(1/√5)
3x = 0.4636 or 2.2143
x = 0.1545 or 0.7381

Answer 18

sin3x = -1/√5
sin^(-1)(sin3x) = sin^(-1)(-1/√5)
3x = -0.4636 or 3.6052
x = -0.1545 or 1.2017

Answer 19

the period is 2pi/3 because the period of sin is 2pi/k.

0.1545 + 2pi/3
0.7381 + 2pi/3
-0.1545 + 2pi/3
1.2017 + 2pi/3

BUT!! when i checked my answer on the calculator the 0.7381 should actually be 0.8926
and that instead it should be
0.8926 + 2pi/3

Answer 20

You're Welcome!

Answer 21

The interval of [0,2pi) actually has 12 answers.

Type into your graphing calculator
y= csc3x

y=5sin3x

Answer 22

So right now I'm stressing because my answer is almost correct EXCEPT for that one part i told you

Answer 23

@lochana
please help

Answer 24

@amyhashimoto77 Are you still working on that problem?

Answer 25

@lochana
Yes, please look at my work above.