> What's a sine, cosine, and tangent? I honestly still don't know, because the words themselves are foreign to me.
They are conversion functions between different fraction-based ways of measuring angles.
You can draw a right triangle for the angle you want to build and you can measure it based on the ratio of any two sides of the triangle.
You can also view the angle as a fraction of a circle. It's up to you decide whether a full circle counts as 360 or 2pi (or 400 or 1 or whatever).
sin/cos/tan and their inverses let you convert between the two. Both are useful, neither is always better. The conversions let you use whichever is easier.
The sine/cosine names don't really make sense in Indo-European languages because they are based on terribly mangled old Arabic. No, they do not come from the Latin word "sinus" = bay or bend. Yes, they probably did affect the direction of the mangling because there was this nice Latin word that looked like it ought to have something to do with it... but they started out as Arabic.
The name of the tangent function comes from the geometric tangent as a line that touches a curve. Tangent comes from a Latin word that means to touch -- hence why they keys on a keyboard are called that in some languages.
If you do some fancy geometric drawing involving a unit circle, a radius, and an angle, then the tangent function naturally appears as the length of a line segment that 1) just touches the perimeter of the circle and 2) is at a right angle to the radius.
How do you remember sin and cos in practice? Draw a unit circle, draw a radius at whatever angle with the x axis that you want. The point where the radius touches the perimeter has the coordinates cos(angle), sin(angle). How do you remember the order? Alphabetically, just like the Baltic states.
Tan is the slope of the radius line: sin(angle)/cos(angle).
How do you remember the fraction for the slope of a line? I use a mnemonic: dydx ("dydex").
The unit circle diagram is nicer because it naturally works with angles bigger than 180 and negative angles. If you look at the unit circle with a radius and the intersection point with the circle, you naturally get a right triangle of the kind your mnemonics apply to.
It also has the advantages of being language/culture blind.
My problem (in this example) isn't understanding, it's memorization. I can understand the concepts. But I have to not only understand the different concepts, I have to understand how they are used, I have to memorize a unique word for each of the three concepts, and I have to remember which word is which concept. It's actually four different mental activities, all of which need to be embedded in my long-term memory, and all for a thing I'll almost never use. Yes, people use mnemonics. But then I have to memorize the mnemonic and what it means, which adds a fifth activity to all this. To add more difficulty, each of these things needs a visual/spacial representation. So I need five different mental activities in five different spatial representations to remember and explain trig ratios.
And this is why it took 8 hours for me to do math homework every night.
They are conversion functions between different fraction-based ways of measuring angles.
You can draw a right triangle for the angle you want to build and you can measure it based on the ratio of any two sides of the triangle.
You can also view the angle as a fraction of a circle. It's up to you decide whether a full circle counts as 360 or 2pi (or 400 or 1 or whatever).
sin/cos/tan and their inverses let you convert between the two. Both are useful, neither is always better. The conversions let you use whichever is easier.
The sine/cosine names don't really make sense in Indo-European languages because they are based on terribly mangled old Arabic. No, they do not come from the Latin word "sinus" = bay or bend. Yes, they probably did affect the direction of the mangling because there was this nice Latin word that looked like it ought to have something to do with it... but they started out as Arabic.
The name of the tangent function comes from the geometric tangent as a line that touches a curve. Tangent comes from a Latin word that means to touch -- hence why they keys on a keyboard are called that in some languages. If you do some fancy geometric drawing involving a unit circle, a radius, and an angle, then the tangent function naturally appears as the length of a line segment that 1) just touches the perimeter of the circle and 2) is at a right angle to the radius.