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!

image

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.

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  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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