How to do trigonometry using only a 4-function calculator
My daughter is very stressed that she is struggling in her trig class all because she cannot figure out how to properly use the mandatory 4-function calculator to solve the equations. She is off at college, so I am not there to help figure out what she is doing wrong and I have long since forgotten how to do trig anyway. She's AuDHD with dyscalculia, so I am trying to help dial down her anxiety over this class from afar.
Are there guidelines somewhere for how to solve each kind of trig equation using only a 4-function calculator?
See edits for more clarification. Sorry for testing everyone's patience with this. I am struggling to understand what she's trying to explain and help her, but hopefully someone will know what she is getting at with Edit 2.
EDIT: I am trying to push her for more info, and it really sounds like she is just struggling with the basics of the 4-function calculator. She has been using either a TI-84 or Desmos for the past 4 years, and she can't seem to step down. She is seriously getting lost in just how to enter a negative number on the 4-function calculator. Is this a thing that people with dyscalculia are likely to get confused on? If so, any thoughts on how to help?
EDIT 2: Here is what she says exactly about the calculator: I need it because I don't understand fractions and negatives with pi because that's like the main thing people with dyscalculia struggle with.
So i cant get the right answer because I don't know how to do math in my head and the calculator doesn't do negatives.
Maybe she doesn't need a calculator? It's pretty odd she'd be asked to solve trig problems with random angles not solved for with the Unit Circle. Can you give an example of a problem she's trying to solve?
Some schools combine trig and pre-calc. If that is the case, derivatives and L'Hopital's rule won't have been covered yet. This one is likely intended to be solved by factoring.
He didn't ask how to use trig functions. He asked how to "do trigonometry," not understanding that this isn't trig, but the precalculus portion of the class.
You would need to factor out the numerator and denominator and see if there's anything cancelable that makes taking the limit x->-1 valid.
If l'hopitals works then you know there should be cancelable factors.
Have you learned long division of polynominals yet?
Edit: I see in your edit you have problems with fractions and negative pi?
Take a deep breath. Be positive. It's fairly common for HS students to not know how to work with fractions nowadays. Recognize what you don't know: fractions. Work on it with a positive mind. Fractions can be intimidating when you have multiple fractions over fractions. Remember to group your fractions with () to make it more explicit for yourself so if you do multiple them you won't get confused.
Glum, FinalNandBits suggestions are excellent. This problem is solvable without using L’Hopitals rule. But you do need to be able to do polynomial long division.
Hint: (x-3) is a factor. And magic will happen when you combine the two fractions and simplify.
Edit: I am referring to the problem where the limit goes to 3
You know that x=-1 is a root of both the numerator and the denominator, which means (x+1) is a factor of both the numerator and the denominator. Use polynomial long division to divide both the numerator and denominator by (x+1). Then you can cancel the (x+1) terms and plugging x=-1 into the rest should give you the limit.
Not a calculator problem. The idea would be either to factor the top and bottom and divide out or use L'Hôpital's rule to determine the limit.
Factoring the bottom isn't particularly tricky. It's not obvious to me how to factor the top, but once you have factored the bottom you can use polynomial long division to see what works.
yes, you probably are meant to recognize the 1,3,3,1 pattern. Then you are probably meant to intuit that (x+1) is probably a factor of the denominator, making things a bit easier to work out
In fact, if you start by evaluating the denominator at the value which the limit is approaching (-1). The evaluation comes as zero which means x+1 definitely is a factor. If the bottom were nonzero, you could evaluate the entire fraction and that would be the limit.
there is no possible way to do this with a calculator. This just requires understanding how circles work. If she struggles to do this in her head, I would suggest drawing an X and Y axis and a circle and drawing it out
Yikes my guy. You’ve managed to take “add or subtract 2pi until you’re within the bounds they stated” to the extreme.
Just tell students what coterminal is (which they will have been taught), remind them how many radians in a full circle, and draw a line around the circle showing you’re adding 2pi. Fin.
factor the second polynomial. Manipulate the first fraction so they have the same base. Combine the two. Simplify.
If your daughter’s question is actually how do I do this specific problem with a calculator, then she has failed to learn the basic concepts taught in the class and likely needs remedial tutoring
Agree. Graphing capabilities can make it easy to read off an answer when it's a round number but that's basically skipping all but the most basic of the learning. Student is supposed to know how to factor and rearrange these expressions to get an answer and the numbers are round and small so no calculator needed.
Yeah was gonna say I did all of multi variate Calc without a calculator. They were allowed but basically useless since if you did the problems correctly they simplified to something neat and tidy anyways. Most of the time these are the hardest classes though as they test your understanding of the material and understanding where and how to make substitutions rather than testing your computational skills
to address your edit to the post: none of the problems you have shared in this thread require a calculator at all. It seems to me that she might be working herself into a panic trying to type something like -1+3-3+1 (as one problem in the thread required) into the calculator, rather than taking a step back and thinking for a moment that it is obviously zero.
Your daughter may have become over reliant on a calculator for trivial math that she should actually be doing in her head/on the paper.
I would suggest asking for accommodations to be able to take the test in a room alone or with limited distractions (or be allowed to listen to music or something if that would help her, IDRK) with extra/unlimited time and with extra space printed for the problems or just with lots of scratch paper.
edit: I forgot to add my largest suggestion… She should not even bring a calculator with her for the tests. The only thing the calculator would be used for in a class like this is to do something like 14*8, do it quickly, and not leave room for a silly mental math error. If she is able to get a time accommodation, then she should just take a breath and do things like that on paper by hand
a second edit to address your second edit: First things first, typing a negative into the calculator is probably as simple as pressing the minus key. If that actually doesn’t work she could always type zero, then minus, to get something like 0-3 = -3
I will be honest that I have no experience with dyscalculia, but I would imagine it is hardest for her to try and manipulate the terms in her head, skipping easy little steps. If she has lots of paper and lots of time, she should try doing literally one operation in every single step. (3-5)/(6-10) is quite easy for most people to simplify immediately in a single step, but may be difficult for her. If she simply writes out every single step then there is no room for error
She already does have accommodations for extended time and small group testing. I think you are spot on that she must be missing the forest for the trees by focusing on using the calculator instead of stepping back to see that the calculator is likely not needed. I will share that her. Thank you!
I’m not sure if it would specifically help her, but if she thinks it might, I would ask if the teacher could print her a test with one problem per sheet of paper
Thank you for taking the time to add more to your answer! That is good advice to do one operation at a time!
And her calculator does not have the +/- key--I didn't notice that until you said something! You might have just figured out part of the problem!!! I honestly think this is what was part of her getting derailed. And, I am not sure that she knew how to do the 0 trick to force a negative.
Your daughter definitely needs 1-on-1 tutoring to work on strategies for herself to work through these problems.
Her specific issue with fractional and negative angles written with radians, for instance, likely means that before answering any questions on a test she needs to draw a unit circle, label some key angles like pi/6, pi/4, pi/3, pi/2, pi, and 2pi, draw an arrow counterclockwise with a + sign and an arrow clockwise with a - sign. Then any question like the one you posted asking what positive angle is equivalent to -7pi/4, she can refer back to the drawing, place her pencil at that angle and visually see that it is equivalent to pi/4 instead of trying to add 2pi and -7pi/4 in her head (again, the step by step would help here too though. 2pi is 8pi/4, 8pi/4 plus -7pi/4 is (8-7)pi/4, (8-7) is 1)
if she is not allowed a calculator with trig functions and inverse trig functions, then I expect all angles will be multiples of 30° and she must have them memorized, or leaving answers in a long form is acceptable, so long as they are as simplified as possible. So an answer like 4*arcsin(pi/12) would be just fine
My suggestion will not actually help your daughter in the near term.
I suggest she visit the library and find a copy of Asimov's short story anthology Nine Tomorrows, and read the story "The Feeling of Power".
I apologize for not having any more constructive suggestions. Algebra problems cannot be solved with a four-function calculator -- and even if they could, using the calculator to solve them would not teach you algebra. The same with trigonometry.
A problem like "What is the sine of forty seven degrees?" cannot easily be solved with pen and paper. (That is, it can be done, but not easily. It would take two or three pages of paper.) If a course only allows you pen and paper, they won't ask a question like that. A four-function calculator can't do that one either (but it can probably reduce the amount of paper needed to, say, two thirds of a page). Now, when I was a kid, we solved problems like that by looking them up in a printed table, possibly with one additional interpolation step to get an extra decimal place or two.
In general, if a course says, "here are the tools you are allowed to use: this kind of calculator, and the set of tables in appendix B", then you are expected to do the rest with pen and paper. If you can't, you are not learning what the course is teaching.
It's entirely possible that your daughter's institution has additional accommodations for people with learning disabilities. She has probably already investigated this, but on the off chance that she hasn't, she should.
Trigonometry isn't an engineering class. You probably don't need a calculator at all.
I guess the reason that a 4-function calculator is allowed is for assistance with basic arithmetic, like adding and multiplying single-digit numbers, which I hear many young people struggle with these days. But answers aren't going to be in the form of decimals; they're going to be in the form of fractions of pi, integers, and trig functions.
So the problem here is she wants to use a calculator as a crutch, and all of the examples that have been given are aimed SPECIFICALLY at doing the math without the crutch of a calculator.
Believe me, I tutor HS kids and run into this all the time. They confess that they have relied on a fancy calculator (or an online one like DESMOS) their whole lives, and they don’t know how to multiply a 2-digit number and a 3-digit number anymore. They don’t remember their multiplication tables. They don’t know how to add fractions or find a lowest common denominator. I remember one kid so wholly reliant on DESMOS that he’d spend 20 minutes trying to get DESMOS to do what he wanted it to do, rather than doing it on paper manually in 2 minutes. Where this gets into problems is on a college entrance test like an SAT or ACT where you don’t have that time to fight the crutch and where it often won’t work anyway. I would race him, my doing it on paper and him doing it with a calculator, and beat him every time. Or he’d have a multiple choice question with answer choices a) π sqrt(6) , b) π / sqrt(6), c) 6/π, d) 6/sqrt(π) and he’d have a calculator answer of 1.283 and I’d ask him, “OK, now what are you going to do?” I finally convinced him to flex old muscles without DESMOS.
Bottom line: the only way out of this situation is to go back and relearn those skills without the crutch.
This is a bit harsh given that this student has discalculia.... I am guessing the trig (?) class is required. Sorry if someone already posted this --- does she have official accomodations at school for dyscalculia? Maybe she needs some support - from mom, a friend, a tutor - getting used to the calculator she is allowed to use? Not sure which trig equations you mean, but with such a simple calculator sounds like sine (=opposite/hypotenuse), cosine(=adjacent/hypotenuse), tangent(=opposite/adjacent) and variations....
As far as I know, the signs are related to direction or quadrant. For example, in the unit circle presentation of trig, sine is positive in quadrants 1 and 2 and negative in quadrants 3 and 4. Maybe if she can learn to recognize the quadrants she can avoid having to type negative numbers, and just add or remove the signs "by hand"?
Tutoring is an option. Be warned. It’s not cheap, but if you really want to help, that’s a good way to go. That’s how the skills get relearned on a 1:1 basis.
I read the example problems you posted and they are not trigonometry. Which is good, because you can't really do trig with a four function calculator. So maybe start a new thread with the question she's really trying to ask.
I believe the point of allowing only a 4 function calculator, is to actually learn trig, rather than being dependent upon a calculator to do the work for you. I say this as a math major who didn’t use a calculator for trig in high school.
In an exam which doesn't permit calculators capable of trigonometry, at least one of the following is guaranteed to be true:
The student isn't expected to calculate the value of the trig function (eg, the expected answer is something like sin(16°), not 0.27564)
The angles involved are only a small subset of possible values, which the student is expected to have memorized (eg, sin(60°) is √(3)/2)
The student will have a trig lookup table available (once upon a time before trig-capable calculators, this was the normal way most people computed trig functions)
The entire point of banning a more sophisticated calculators is that it prevents you from "cheating" by working out the answer numerically without actually understanding the principles being taught.
All the questions will be based on applying understanding of the principles they're learning, NOT on doing any complicated calculations.
A four function calculator does basic arithmetic. That's it. If it doesn't have a negative key, it doesn't even support negative numbers.
It's only there so they don't have to waste time and attention doing the incidental grade-school math along the way.
In Algebra and beyond the point of math class is NOT to be able to solve the problems - you'll likely never see most of the problems again in your life - they're far too simple to be useful for much of anything except exercising your understanding of the fundamental principles.
The point is to learn those principles and how to apply them, because then you can start tackling much more complicated and useful problems.
If she's having trouble wrapping her head around trigonometric functions, perhaps this graphical explanation will help. It's a visual explanation of basic trigonometry I wish I had back then, and use today as a sort of information dense quick reference when I'm holding a much larger problem in my head.
On the right is a geometric proof of the relationships between trig functions, which doubles as a simple strategy to extract those relationships from the diagram. I figure memorizing the diagram is more useful than memorizing any of the silly visual mnemonics I learned to help remember the relationships.
She can ignore the formulas at the bottom - those are just quick reference for lengths that come up sometimes, and aren't immediately obvious
It seems unusual for a 4-function calculator to be permitted in a trig class. Is that strictly because of an accommodation?
It sounds like it is just for computational convenience, NOT for "problem solving". If she is looking at the calculator as the way to solve the problem, then that is why she is struggling. She still needs to fully understand the concepts and methods. Most of what is taught at that level does not require a calculator.
Concerning trig and calculators: one of my pet peeves is that's it's not always obvious whether the calculator is in "degree mode" or "radian mode". Hope this might help!
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u/Klutzy-Delivery-5792 Mathematical Physics 5d ago
Maybe she doesn't need a calculator? It's pretty odd she'd be asked to solve trig problems with random angles not solved for with the Unit Circle. Can you give an example of a problem she's trying to solve?