Question

(2 sin2 x − 1)(3 tan2 x − 1) = 0

Solve the equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)

Answer 1

I entered the answer as pi/4, -pi/4, pi/6, -pi/6 but it was marked wrong

Answer 2

@Please.help.me
Did the question limit the answer to [0,2\(\pi\)) like the other problem?
If not, do you know what you need to do?

Answer 3

The only directions were "Solve the equation. (Enter your answers as a comma-separated list. Use n as an integer constant. Enter your response in radians.)"

Answer 4

So you're confirming that there is no restricted interval for the answer, which means that we need to give ALL the solutions.
Do you know the periods of the sine and tangent functions?

Answer 5

2pi?

Answer 6

and pi?

Answer 7

Exactly!
Which means that if \(\theta\) is a solution for a sine function, automatically \(\theta\)+2n\(\pi\) is a solution, where n is an integer.
Same goes for tangent.
Can you post your improved solutions here?

Answer 8

I still do not understand how I can get the right answer by knowing the periods?

Answer 9

I forgot to note that your answers are correct for 0-2pi, but not if there is no limited interval.

Answer 10

So there are more solutions?

Answer 11

So you will need to extend your (correct) answers to all possible solutions, in other periods, like x+2pi, x+4pi, x+6pi, ..... and the way to write that is x+2n pi where n represents ANY integer.

Answer 12

\[2 \sin ^2x-1=0\]
\[or 1-2\sin^2x=0\]
cos 2x=0\[\cos 2x=0=\cos \left( 2n+1 \right)\frac{ \pi }{ 2 },where~ n~ is~ an~ integer.\]
\[2x=\left( 2n+1 \right)\frac{ \pi }{ 2 },x=\left( 2n+1 \right)\frac{ \pi }{ 4 },n=0,1,2,3\]

Answer 13

Actually, n can represent ANY integer, positive or negative. But that has been defined for you by the question.

Answer 14

So if I type pi/4, 3pi/4, -pi/4, 5pi/4, pi/6, -pi/6, 5pi/6

Will it be marked right?

Answer 15

Based on the instructions given

Answer 16

It will not be correct, because you are missing MANY solutions, for example
-pi/4 is a solution, -9pi/4 is also a solution, -17pi/4 is a solution, you can write for the next two years, and you still haven't given all the possible solutions.
However, if you write pi/4+2n pi, that would represent all possible solutions (infinitely many of them). n has been defined by the question as any integer.
But you need to repeat that four times, once for each of your solutions, and make sure you take into account of the period of the functions (sine: 2pi, tangent: pi).

Answer 17

Also thank you so much for taking the time to help me with this problem!

Answer 18

Is everything ok so far?

Answer 19

If so, you can post the revised solution.
Also, -pi/4 (solution for sine function) can be rewritten as 2pi-pi/4=7pi/4.
And -pi/6 (solution for tangent function) can be rewritten as pi-pi/6=5pi/6.
When we use the n (integer), we prefer to write the basic solution within 0-2pi.

Answer 20

How many solutions are there supposed to be in total?

Answer 21

Nice presentation!

Answer 22

\[3 \tan ^2x=1,3 \tan ^2x=1,\tan x=\pm \frac{ 1 }{ \sqrt{3} }\]
\[\tan x=\frac{ 1 }{ \sqrt{3} }=\tan \frac{ \pi }{ 6 },\tan \left( \pi+\frac{ \pi }{ 6 } \right)\]
\[x=\frac{ \pi }{ 6 },\frac{ 7 \pi }{ 6 }\]
\[\tan x=-\frac{ 1 }{ \sqrt{3} }=-\tan \frac{ \pi }{ 6 }=\tan \left( \pi-\frac{ \pi }{ 6 } \right),\tan \left( 2 \pi-\frac{ \pi }{ 6 } \right)\]
\[x=\frac{ 5 \pi }{ 6 },\frac{ 11 \pi }{ 6 }\]

Answer 23

That answer was marked wrong for some reason

Answer 24

There are four basic solutions (within 0-2pi), but infinite in total when we take into account of all possible values of n (infinite).

Answer 25

Sorry, actually, pi/4, 9pi/4 are within 0-2pi, and pi/6, and 5pi/6 are between 0 and pi.

Answer 26

So those are the final solutions?

Answer 27

As I mentioned earlier, EACH of the four basic solutions must be rewritten in terms of n.
For example,
pi/6 must be rewritten as pi/6+n pi to take into account of the solution in all other periods (of tangent).

Answer 28

pi/4+n pi, 9pi/4 +n pi, pi/6+n pi, 5pi/6 +n pi ?

Answer 29

This is an example that is written right. So would that form work too? If this is an example of that problem?

Answer 30

(It is in the screenshot)

Answer 31

pi/4+n pi, 7pi/4 +n pi, pi/6+n pi, 5pi/6 +n pi
yes, that's it, but I probably made a mistake, it's 7pi/4, not 9pi/4.
AND, don't forget, solutions for tangent have a period of pi, and solutions for sine have a period of 2pi. You need to adjust that.

Answer 32

Did you take a look at the screenshot?

Answer 33

I just did, the only difference is that they write the n pi term before. The usual way is to write it after, so the basic solution comes first. I guess it is easier to programme the other way, but computer should accept both.
Remember, solutions to sin(x) function has a period of 2pi, and solutions to tan(x) functions have a period of pi.

Do you understand why we need to do this period thing?
Look at the graphs of sine and tangent in the following link:
https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html
and draw a horizontal line at +sqrt(2), and see which points the line cuts the sin(x) function. It would help understand the concept.

Answer 34

pi/4+n 2pi, 7pi/4 +n 2pi, pi/6+n pi, 5pi/6 +n pi ?

Answer 35

How do I write it the other way to make sure the computer accepts it before I get it wrong?

Answer 36

Actually, looking at the graph, we have missed some solutions for the sine function.
The basic solutions for the sine functions between 0 and 2pi are
pi/4, 3pi/4, 5pi/4 and 7pi/4.
You can see this by drawing lines y=sqrt(2) and y=-sqrt(2) on the sine curve.

Answer 37

So these pi/4, 3pi/4, 5pi/4 and 7pi/4 are the final final answers?

How do I write them exactly like the example?

Answer 38

No, I take it back. The four above (from sine) can be reduced to two by making the period to n pi.
The format would be like
n\(\pi\)+\(\pi\)/4
n\(\pi\)+3\(\pi\)/4
n\(\pi\)+\(\pi\)/6, and
n\(\pi\)+5\(\pi\)/6

Did you take a look at the graphs of sine and tangent?

Answer 39

Yes, I did thank you! So these are now the final solutions to the problem?

Answer 40

Yes, as far as I can see.
I suggest you take a look at the two posts @sshayer kindly posted. They should represent the same solution that we got.
If they match ours, then you'd be sure they are correct.

Answer 41

Yes! Thank you so much!!! Sorry for asking so much questions and taking so long to understand!

Answer 42

You're welcome. But I still think you want to understand why we are adding an infinite number of solution (using n) by studying the graph of sine, cosine and tangent.
Understanding the concept will help you solve other problems.

Answer 43

Yes, I need to read over everything again and study it because these problems are very new to me

Answer 44

Yes, please do. Also study what @sshayer wrote. They are formal solution, what you would write in a written exam.