Hard enrichment question from the Cambridge textbook! (1 Viewer)

[catha230]

Could someone please help me with the following question? I have no idea how to approach it. Thank you!

 

[fan96]

In more concrete words, you are asked to show



given the conditions





So, try to solve for . You have three equations with as many variables, so intuitively this should be solvable.

Bonus points - this question is not quite correct, because there are other values of that work too. Can you find them?

This is a possible (but long-winded) solution. Probably not the intended one?

Spoiler: Solution

From the last two equations, , so we have (by the general solutions to trig equations)



or for some integer .

So either or .

The former implies that , so it cannot be true.

Looking at the first two equations,



so

.

Now changing can only flip the sign of the LHS, since . Because we really only care about , we can just pick for simplicity. Make the substitution to get

.

We can expand the LHS using angle sum formulae. Then use to write everything in powers of :



Now cannot be true, otherwise would be an odd multiple of and so .

Thus and so

.

for any integer .

The positive case with is numerically equivalent to . Put these in a calculator to convince yourself this is true (or even better, try proving it).

At this point we're done, but if you want to graph these results you can solve for by using .