Now that we know where the sine and cosine functions came from, we can start to connect all these triangle measurements to angles. And from there, we can move towards circles and then, finally, that long and hilly wave function. But that’s a ways away. So first things first.
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Sine and Cosine Revisited
We know the definition of sine and cosine:
sine angle (∠) B = opposite / hypotenuse = b/c
cosine angle (∠) B = adjacent / hypotenuse = a/c
As we learned, sine and cosine aren’t about specific triangles. They’re ratios based on right triangles. That ratio only depends on the angle. Once we have the angle, we can create a right triangle at any point along the hypotenuse and the ratio will be the same. The triangles will be proportional, like I showed before.
sine (∠B) = b/c = b’/c’
cosine (∠B) = a/c = a’/c’
This is where the math turns from triangles to angles and we start shortening sine and cosine to sin and cos. Because as long as we have a right triangle, and as long as that angle B is the same, then the sine and cosine will be the same. For mathematicians, this might be a simple idea, but for a lounge sitter turned librarian, this was HUGE. Almost an epiphany.
It’s all about B. We just need the flat side and a straight up side and we can get the ratios for the trig functions.
This is how we can from from sine and cosine to their abbreviations sin() and cos(). These are called functions, because some number, or some variable, goes in the middle of the parenthesis. From now on, for sin() and cos(), what’s in inside those parenthesis will be some kind of angle measurement.
Axes
What mathematicians do to make calculations easier is to put the triangle on a set of axes. They put the flat side, which would be ‘a’, along the x axis and shoot the angle line out from the (0,0) spot (the origin), like in the diagram below.
We (I mean ‘the mathematicians’) create the angle B using the x axis as the flat side, then a line parallel to the y axis as the up side. From that you have all your trig functions. The math isn’t easy, and we’re not going to go through it all, but it is possible. That’s how calculators do it. They use something called the CORDIC algorithm to make the calculations. (Wikipedia explanation)
Triangles to Angles
Knowing that for any angle, we can create a right triangle from which we can calculate the sine and cosine, we can now drop the triangles and go only with the angles.
>90°
In the previous math post, we went through the quadrants and the calculations for the sine and cosine in those different quadrants. We used triangles to show when they would be positive and negative. Now that we’re moving on from triangles, we’re just going to use the angles, but it’s still the same thing.
Conventions:
A positive angle is formed by moving the angle-line counter-clockwise (up) from the zero line.
A negative angle is formed by moving the angle-line clockwise (down) from the zero line.
It’s important to remember that the standard starting point for measurement is flat along the x axis, pointing to the right. That’s 0 degrees. One full rotation around the set of axes equals 360 degrees. Half the rotation, to the opposite horizontal line, equal 180 degrees. Moving the angle-line up is moving in the positive direction (a positive angle is created) Moving it down is moving it in the negative direction. (a negative angle is created.) There is nothing magical about these numbers It’s just mathematical convention.
In the diagram above are two examples. In the first one, in quadrant two, we’re showing that the sine of a 110o angle is equal to the sine of a 70 degree angle. The cosine, however, is going to be the negative cosine of a 70 degree angle. And if you’re using a quadrant three angle, like a 225 degree one, we can create a right triangle in the 3rd quadrant with a 45 degree angle to the x axis. In this case though, both b and a will be negative, and the sine and cosine will both be negative as a result.
In this way, it’s easy to see that for any angle-line attached to the origin, we can create a triangle and come up with the appropriate sine and cosine.
Sine 110 = sine 70 = 0.94
Cosine 110 = – cosine 70 = -0.34
Sine 225 = – sine 45 = -0.71
Cosine 225 = – cosine 45 = -0.71
For every angle from 0 – 360, a sine and cosine value exists and can be calculated.
>360°
What do we do about a 500° angle, or even a -420 degree angle? They exist, but what do they mean to sin and cos. As pictured below, moving the line more than one rotation creates a 360o angle, either positive or negative.
As in the pictures, the sin(395°) = sin(35°). All of the trig function calculations are the same for a 395° angle as they are for the 35o angle. For negative angles, it works the same, we’re just spinning the line in the opposite direction. The sin(-395°) = sin(-35o).
No matter how many spins are made, the trig functions will be the same.
Next Up, Circles
Now that we’ve got the sine and cosine reduced to sin() and cos(), and have a good idea how they have a value for any angle, no matter how big or whether it’s positive or negative, the next is to figure out how circles come in, because they do, and so does this thing called a radian. But that’s coming up.