The right order for the fundamental trig identity

If you google “fundamental trig identity” you will get many many images and handouts which all list the fundamental trig identity as:

sin2 t + cos2 t = 1

This is, of course in the wrong orderĀ and it should really have cos firstĀ then sin, like this:

(cos t)2 + (sin t)2 = 1

“But David,” you say, “it’s addition, so it doesn’t really matter what order it’s in does it?” Of course it does! Mathematically it’s the same, but psychologically it’s different. If it really wasn’t different then you would sometimes write cos first and sometimes write sin first, but I can bet you always write it in a particular order. And if you write it with sin first, then you’re making it harder for yourself.

Let me explain.

The reason we have the fundamental trig identity is because the angle t there is a piece of the circumference of a unit circle, and cos t and sin t are the coordinates of the points on that unit circle. If I asked you to write down an x-y equation for the unit circle, you would naturally write x2 + y2 = 1 with the x first. But the x-coordinate of a point on the unit circle is cos t, and the y-coordinate is sin t, so of course that means it’s (cos t)2 + (sin t)2 = 1. Writing your trig identity with the cos first makes it easier to make the connection with the equation of the unit circle. If you write it with sin first you’ll have to continually switch it round!

Also, the order does matter if you’re using hyperbolic trigonometry. Then the formula is (cosh t)2 – (sinh t)2 = 1 and having sin first would be definitely mathematically wrong. For years, I had great trouble remembering which way around this was supposed to go until I realised that the cos and sin were in alphabetical order. From that point forward I always wrote my ordinary trig identity in the same order as the hyperbolic trig identity (in alphabetical order) so that through force of habit I would never get the hyperbolic one wrong.

So, I recommend you start writing your fundamental trig identity in the right order. It might help you remember and make connections to other things!

PS: You may be wondering about the way I put brackets in my trig identities. Don’t even get me started.

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