Trig questions

amathproblemthatneedsolve

amathproblemthatneedsolve

Junior Member
Joined
May 12, 2019
Messages
189
I have a few trig questions below that I am wanting to make sure I am answering correctly, and need help finishing some of them.

  1. Sketch the following graphs for the given domain.
  2. y = sin x 0 ≤ x ≤ 4 π
Is this graph what is required for the problem? https://www.desmos.com/calculator/sreqdukoid

Solve the following equations for the given domain.
  1. tan x = 1.9 0° ≤ x ≤ 360°
  2. sin 2x – 3 = –2.4 0° ≤ x ≤ 360°
  3. 5 cos 12 x = 0.5 0 ≤ x ≤ 2 π
  4. cos (2x + π) = –0.2 0 ≤ x ≤ 2 π
1. arctan(1.9) = 62.2414594 so x=62.24° x=360°-62.24° = 297.76degrees. So, solutions are 62.24 ° and 297.76°
2. Sin 2x = 0.6 apparently(maybe they made an error). x = 15.48°. But I'm getting x=17.46degrees
3. cos (12x)= 0.1 x= 84.26degrees . Is there a second solution?
4. 2x+π = 101.54° I don't know how to get x from here next step? I believe the answer is x = -39.23°
 
Last edited:
You should use more punctuation in your work so that it is more readable, and use words to explain what you are doing.

But the graph is correct (though I'd show only the part you shaded in).

For (1), why did you subtract from 360°? And did you check your answer?

For (2), I get neither of the answers you give; and there should be four answers. Are you saying the book gave 15.48° as the only solution?

For (3), you seem to have ignored the 12, and the fact that the answer should be in radians, not degrees. And, yes, there are others.

For (4), use radians and it may make more sense to you.
 
Dr.Peterson said:
You should use more punctuation in your work so that it is more readable, and use words to explain what you are doing.

But the graph is correct (though I'd show only the part you shaded in).

For (1), why did you subtract from 360°? And did you check your answer?

For (2), I get neither of the answers you give; and there should be four answers. Are you saying the book gave 15.48° as the only solution?

For (3), you seem to have ignored the 12, and the fact that the answer should be in radians, not degrees. And, yes, there are others.

For (4), use radians and it may make more sense to you.
Click to expand...
Sorry, for the poor punctuation. An example problem was given as well as a solution:
tan x = 2.5 , 0° ≤ x ≤ 360°
x = 68.2° (calculator)
x = 180º + 68.2º (symmetry)
= 248.2º
So, solutions are 68.2° and 248.2° (1 d.p.



So for question (1), I will adopt the same method.
tan x = 1.9, 0° ≤ x ≤ 360°
x= 62.24° (calculator)
x=180° +62.24°
=242.74°
So, solutions are 62° .24 and 242.74°
Although I used the same method I don't know whether it was correct to or whether it resulted in the correct answer?


(2) sin 2x – 3 = –2.4, 0° ≤ x ≤ 360°
Not sure here

I made an error here it's 1/2, not 12
(3) 5 cos 1/2 x = 0.5, 0 ≤ x ≤ 2 π
dividing both sides by 5 gives me
cos 1/2x=0.1
1/2x= 1.4706
1/2x = 1.4706, 2π -1.4706, 2π + 1.4706, 4π – 1.4706
1/2x = 1.4706, 4.8125, 7.7537, 11.0957
x= 2.9412, 9.625,15.5074,22.1914


(4) cos (2x + π) = –0.2, 0 ≤ x ≤ 2 π
Doing inverse cos(-0.2)
(2x + π)=1.772154248
1.772154248+π= 4.913746901
2x= 4.913746901
x= 2.456873451
π-2.457=0.685,π+2.457=5.598,2π-2.457 = 3.8261
 
(1) Do you understand WHY you should add 180 rather than subtract from 360? That is the important thing! And the other important thing is, do you realize that you can check whether your answer is correct (at least in part) without asking?

(2) Try doing it again, and show your work this time. It may be a silly mistake with the calculator or something.

(3) Looks good, but check which solutions are in the domain, and then make sure they are solutions by putting them into the original equation.

(4) Why did you add pi? And you should find the other inverse cosines immediately, not at the end. What you did at the end applies not to x, but to 2x + pi. Do you understand that?
 
Dr.Peterson said:
(1) Do you understand WHY you should add 180 rather than subtract from 360? That is the important thing! And the other important thing is, do you realize that you can check whether your answer is correct (at least in part) without asking?

(2) Try doing it again, and show your work this time. It may be a silly mistake with the calculator or something.

(3) Looks good, but check which solutions are in the domain, and then make sure they are solutions by putting them into the original equation.

(4) Why did you add pi? And you should find the other inverse cosines immediately, not at the end. What you did at the end applies not to x, but to 2x + pi. Do you understand that?
Click to expand...

(1) To be quite honest I don't know why. I know that I can graph it and check where the points intersect to see if I got the correct results.

(2) sin 2x – 3 = –2.4, 0° ≤ x ≤ 360°
sin 2x= 0.6
2x= 36.8689 I know there is 3 more that I am meant to get at this stage but unsure how.
x=18.435
(other x= values 71.565degrees,198.4349degrees and 251.5650degrees)


(3) The only solution which is in the domain is 2.9412

(4) cos (2x + π) = –0.2, 0 ≤ x ≤ 2 π
2x + π= 1.772154248
[MATH]x=\frac{\pi +1.77215\dots }{2},\:x=\frac{\pi -1.77215\dots }{2},\:x=5.59846\dots ,\:x=3.82631[/MATH]
 
(1) The proper check is not to graph (which needs technology to do well), but to plug the answer into the original equation and see that it is true. (Of course, graphing also shows how many solutions there are, in case you missed any, so it is also worth doing.)

In the example solution you quoted, they briefly stated the reason: symmetry. In this case, the "symmetry" is actually the periodicity: the tangent repeats every 180 degrees. I tend to think of it in terms of the unit circle definition; two points on opposite sides of the circle have the same tangent, because the tangent is the slope of the line.

(2) At the point where you say 2x= 36.8689 you need to observe that the other angles with the same sine are the supplement of this angle (pi - 36.8689), and any angle coterminal with either of those: 36.8689 + 2 n pi and (pi - 36.8689) + 2 n pi, for any integer n. When you finish the solution, you have to decide which values of n give you a value in the required interval. You've been doing this, just too late.

Note that this time you got the number I got. It may be helpful for you to figure out where you got the two other answers you gave before.

(3) Good.

(4) Don't you see the little error? Be sure to think! When you solve 2x + π= 1.772154248, you have to subtract π, not add it!

Frankly, we are demonstrating why we ask everyone to submit only one problem at a time. Trying to juggle four problems at once, which share some of the same issues, makes me groan when I see you've answered and I have to deal with all this again when I have other things to do. Please, next time, follow the rules.
 
Thanks for your help. I have one last question on how you suggested I checked them. I am encountering a problem with sin 2x-3 = -2.4 When I substitute my values into x I am not getting it to =-2.4 why would this be? By checking on a graph I can clearly see the values are correct. ( Ps my values for the equation are 18.435, 71.565,198.435,251.565 which are all in degrees) Note my calculator is in the correct mode also.
 
amathproblemthatneedsolve said:
Thanks for your help. I have one last question on how you suggested I checked them. I am encountering a problem with sin 2x-3 = -2.4 When I substitute my values into x I am not getting it to =-2.4 why would this be? By checking on a graph I can clearly see the values are correct. ( Ps my values for the equation are 18.435, 71.565,198.435,251.565 which are all in degrees) Note my calculator is in the correct mode also.
Click to expand...
@amathproblemthatneedsolve, do you think that the sine notation is a function? If so why do you post sin 2x-3 = -2.4?
Should you not use function notation \(\sin (2x-3) = -2.4~?\)
 
Sine, Cosine and Tangent are trigonometric functions. That was an error. Plus on a calculator that error doesn't regularly occur since the sin function gets entered as so.... sin().
 
amathproblemthatneedsolve said:
Thanks for your help. I have one last question on how you suggested I checked them. I am encountering a problem with sin 2x-3 = -2.4 When I substitute my values into x I am not getting it to =-2.4 why would this be? By checking on a graph I can clearly see the values are correct. ( Ps my values for the equation are 18.435, 71.565,198.435,251.565 which are all in degrees) Note my calculator is in the correct mode also.
Click to expand...
First, note that pka thinks from the point of view of a mathematician, to whom the sine function is inherently a function of numbers that are equivalent to radians. In real life, degrees are still very important, and your book uses both. A result is that you have to observe carefully which unit is being used. You have done so.

But he also brought up another issue without making it explicit. When you write sin 2x - 3 = -2.4, it is not clear what is the argument of the sine. Traditionally, it means sin(2x) - 3 = -2.4; and if you read it as sin(2x - 3) = -2.4, it would have no solution, so I have been assuming it means the former, as you took it in your work. But now you say you are having trouble checking it. You didn't say what result you did get, which would have been very helpful! But it is possible that you are evaluating using the wrong meaning. If you evaluate sin(2*18.435 - 3), you will get 0.55731. If you evaluate sin(2*18.435) - 3, you get -2.39999857, which rounds to -2.4.
 
Dr.Peterson said:
First, note that pka thinks from the point of view of a mathematician, to whom the sine function is inherently a function of numbers that are equivalent to radians. In real life, degrees are still very important, and your book uses both. A result is that you have to observe carefully which unit is being used. You have done so.

But he also brought up another issue without making it explicit. When you write sin 2x - 3 = -2.4, it is not clear what is the argument of the sine. Traditionally, it means sin(2x) - 3 = -2.4; and if you read it as sin(2x - 3) = -2.4, it would have no solution, so I have been assuming it means the former, as you took it in your work. But now you say you are having trouble checking it. You didn't say what result you did get, which would have been very helpful! But it is possible that you are evaluating using the wrong meaning. If you evaluate sin(2*18.435 - 3), you will get 0.55731. If you evaluate sin(2*18.435) - 3, you get -2.39999857, which rounds to -2.4.
Click to expand...
Right, I was doing sin(2*18.435 - 3), and getting 0.55731. Thanks!
 
amathproblemthatneedsolve said:
1. arctan(1.9) = 62.2414594 so x=62.24°
Click to expand...
I am a curious person. If you concluded that x = arctan(1.9) = 62.2414594° then why would you then say that x = 62.24°???

What I am really saying is that once you find out what x equals then why change it?

I am not saying that you will make this mistake but it is similar.
I have seen student after student say that sqrt(16) = 4 BUT they write down something different. Even though they say (and I hear them say it) sqrt(16) = 4 then write down sqrt(16) = sqrt(4). If you think that sqrt(16) is 4 then write down sqrt(16)=4. If you think that sqrt(16) is sqrt(4) then write down sqrt(16) = sqrt(4). This last equation is wrong but at least it says what you believe the sqrt(16) equals.
 
Last edited:
Jomo said:
I am a curious person. If you concluded that x = arctan(1.9) = 62.2414594° then why would you then say that x = 62.24°???

What I am really saying is that once you find out what x equals then why change it?

I am not saying that you will make this mistake but it is similar.
I have seen student after student say that sqrt(16) = 4 BUT they write down something different. Even though they say (and I hear them say it) sqrt(16) = 4 then write down sqrt(16) = sqrt(4). If you think that sqrt(16) is 4 then write down sqrt(16)=4. If you think that sqrt(16) is sqrt(4) then write down sqrt(16) = sqrt(4). This last equation is wrong but at least it says what you believe the sqrt(16) equals.
Click to expand...
It was meant to be 62.2415° I'm mean to do (4.s.f)
 
amathproblemthatneedsolve said:
It was meant to be 62.2415° I'm mean to do (4.s.f)
Click to expand...

The fact is (and I think this may be what Jomo is saying), even "arctan(1.9) = 62.2414594°" is incorrect, as any rounding changes the value, and the calculator has already rounded in displaying this value, even if you did not. What you should say is "arctan(1.9) ≈ 62.2414594°".

If the instructions specified significant figures, then you should have mentioned that; but even so, (a) 62.2415° has 6 s.f., not 4; and (b) you should use more s.f. in intermediate results than are required for the final result.
 
You must log in or register to reply here.
Top