Calculator not giving exact answers

Hello, for some questions regarding complex numbers, my calculator (casio fx-991EX) sometimes will only give decimals and not any exact values with the surds.

For example when I do root40(cos(arctan(1/6)+2pi/3)) I get just a decimal when I need a surd as it specifies I need an exact answer

In the mark scheme it says this is the correct thing to do but how am I meant to get an exact answer if the calculator doesn't give me it.

Press d <=> s

I can't comment on your specific calculations above but Sometimes with calculators to keep an answer 'exact' you have to put down your calculator and use your skills with surds and constants like pi to manipulate the calculation and produce the required simplified results.
Even something as simple as pi x root(2) will only display as a decimal on most scientific calculators.
Another common issue with not getting exact answers in trig problems is whether you are setup to use radians or degre

Maybe just leave it in the original form then. If that's the penultimate step before your weird decimal, then maybe that is the simplest form

It probably wants you to figure out the exact answer without the calculator.

Your calculator is going to mostly do stuff like cos(arctan(2)) in a purely numerical way, it won't calculate it like a human would with something like 1 + tan^2 = sec^2. (with this identity you can show that cos(arctan(2)) = 1/sqrt(5)) I don't know enough about how calculators are programmed to say why it'll sometimes work and sometimes won't. It should give you the symbolic answer if the arc(...) is a rational multiple of pi with low numerator/denominator.

You will need to work it out yourself and check that the decimal expansions coincide.

I don't see how I could work it out without the use of a calculator

use addition formulae for cos then the identity I gave.

It might not be presented in this way but finding cos theta given tan theta, etc., is something you are expected to do in the maths A-level.

Yes I know but that would take far too long for a few marks and I doubt I would end up with an exact value at the end and the mark scheme would certainly mention the use of that identity if it was necessary

i don't think it's particularly long, you can do most of the work on your calculator

have you tried expanding then seeing if your calculator would've given you cos(arctan(1/6))? If so then I don't think it's an unreasonable expectation at all.

i don't think it's particularly long, you can do most of the work on your calculator

have you tried expanding then seeing if your calculator would've given you cos(arctan(1/6))? If so then I don't think it's an unreasonable expectation at all.

Here I have some workings for this, It takes like a page of working for me and this is only the first part of it, I would have to do the same thing for sin,

How is this not time consuming for a 4 mark question? Especially as this is just the last part of the question

the value for cosine is immediate since arctan(2/6) is an acute angle.

Dunno what to tell you. If your calculator can't do cos(arctan(2/6)) then that's what you've got to do.

The problem isnt finding a value for arctan(2/6), it is finding an exact value for root 40 * cos((arctan2/6) + 2pi/3)

My calculator can do cos(arctan(2/6)) but doesn't give an exact value which is the problem. It is obviously not how you are meant to do the question as this is not in the mark scheme at all, the marks do not reflect the work, and I've never seen an a level maths question that has asked to do cos(artcan(x)) before it is just highly unlikely

i know, but since arctan(2/6) is acute you just have to use sin^2 + cos^2 = 1.

Was it an actual exam question or some kind of specimen material? If the latter it's very possible they assumed your calculator can do it.

You can do it geometrically like mqb said but it's the same calculations just with a different presentation. As I said in mechanics I don't think having to find sin theta given tan theta is an uncommon situation.

I didn't think of drawing a triangle, but it is still just too much work as it is the final part of the question (and it has to be done again because the question asks to find 2 points on an argand diagram). It was an exam question from 2019 I believe from a pure paper. Thinking about it now I guess the cos(arctanx) thing can come up, I've never seen it in a pure context where you have to use 1 + tan^2 = sec^2 to find the answer though.