trigonometric identities double angle formulas

0201Ĭos(pi/8) is equal to + or – the square root of 1/2 ((1 + cos(pi/4)), the X is pi/4 and then we are finding the cos(x/2) here. Now let us figure out the cos(pi/8), I will do that on the new page here. So, sin(pi/8) then must be equal to the + square root of (2 – root 2/2). I know that I want to take the positive square root here in quadrant 1, sin(pi/8) must be positive. 0140ĭefinitely in the first quadrant there, both sin and cos will be going to be positive, remember sin and cos, and the x and y coordinates. Let me draw a quick unit circle here, I know that pi/4 is over here, pi/8 is half of that so it is down here. Now, I need to deal with this + or -, figure out whether it is positive or negative. I’m going to put this over common denominator, so ½ is 2/2, (2 – root 2/2), (+ or – the square root of 2 – root 2/4), (+ or – the square root of 2 – root 2), the square root of 4 is just 2 there. Now, cos(pi/4) is a common value that I remember, I know that by heart it is square root of 2/2. Sin(pi/8) is equal to + or – the square root of ½, of 1 – cos(pi/4), because my X is pi/4. I’m going to take the X to be pi/4 and I’m going to plug into the half angle formulas. 0052Īnd pi/4 is a common value, I know the sin and cos of pi/4, that is my starting point.

I do not remember the sin and cos of pi/8, what I do know is that pi/8 is ½ of pi/4. In this case, we want to find the sin and cos of pi/8, now that is not one of the common values that we need to memorize. Same formula just for the plus, square root of 1/2, 1 + cos X. Let me remind you of the half angle formulas, we have sin(1/2x) is equal to + or – the square root of ½, 1 – cos X, we will be using that and cos(1/2x).

Now, we are going to find the sin and cos of pi/8 and we are going to check that our answer satisfies the pythagorean identity sin 2 + cos 2 = 1. Ok we are going to try some examples of the half angle formulas.

Trigonometric identities double angle formulas