To all those studying trigonometry — let’s take the mystery out of special right triangles, shall we? In a trig class, students are expected to know the sine and cosine of angles that are 45 degrees, or perhaps a multiple of 45 degrees. Here is my effort to make it as simple as possible to “see” what these values should be, without having to resort to memorization.

We start with a right triangle with a 45 degree angle. For convenience, let the hypotenuse have length 1. We are dying to find out how long the legs are. To find out, we apply the Pythagorean Theorem, and we get this:

Next we go ahead and square things, giving this:

Time to get clever: since one of the acute angles is 45 degrees, the other one must be as well (how else can you get 180 degrees?). And a triangle with two equal angles must have two equal sides, so — guess what? — the legs of the triangle are equal. And the value of “?” that satisfies ? + ? = 1 is clearly 1/2.

Therefore, in a right triangle with a 45 degree angle, if the hypotenuse is 1, then the legs are both the square root of 1/2. That’s it! Once you’ve done these mental steps a few times, it should be fairly easy to close your eyes and come up with the solution in a matter of seconds — and there is really no need to memorize anything! Just apply the Pythagorean Theorem, and it’s very quick to see that you need to get 1/2 + 1/2 = 1 once you’re done squaring things.

Next post: analyzing 30-60-90 triangles, with no memorization!

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[…] The vertical and horizontal legs must both be sqrt(1/2). Proof: if you square each side in the triangle, you get (1/2) + (1/2) = 1, which is obviously correct. A more detailed treatment of this type of triangle is found here. […]