Home > #MTBoS, A Level, Starters > A lovely simple trigonometry puzzle

A lovely simple trigonometry puzzle

Sometimes a puzzle can look complicated,  but be rather simple (see this geometry puzzle). I love puzzles like this and I particularly like to test them out on classes to try and build their problem solving ability.

Just now, I saw the following trig puzzle from brilliant.org and I love it! It’s amazing!


Have you done it yet?

How long did it take you to spot it?

My initial thought was, it’s got three terms,  it’s bound to be a disguised quadratic that will factorise. A few seconds later I realised that it wasn’t. I saw the – sin^4 and suspected a difference of two squares but then a few seconds later it became clear.

If you haven’t spotted it yet, have a look at the expression rearranged:

Sin^6 + sin^4 cos^2 – sin^4

See it now? What if I rewrite it as:

Sin^4 sin^2 + sin^4 cos^2 – sin^4

I’m sure you have seen it now, but to be complete,  take the common factor of the first two terms:

Sin^4 (sin^2 + cos^2) – sin^4

Obviously sin^2 + cos^2 = 1, so we’re left with:

Sin^4 – sin^4 = 0

A lovely, satisfying, simple answer to a little brain teaser. Hope you liked it as much as I did.

Cross-posted to Betterqs here.

  1. May 8, 2016 at 10:29 pm

    Neat little identity. I saw it slightly differently by changing the given cos squared to 1 – sin squared. End result is the same of course 🙂

    • May 8, 2016 at 10:33 pm

      Aye, somebody else just said that on Facebook too. Both nice solutions. Lovely little task for y12 and 13 tomorrow I think.

  2. flyingcoloursmaths
    May 9, 2016 at 4:52 pm

    Factorises immediately as sin^4 (x) [ sin^2(x) – 1 + cos^2(x) ] = 0.

    • May 9, 2016 at 4:53 pm

      Lovely concise solution as always Colin!

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