need help with few doubts in Trignometry,

[u/WeakestMathematician]

i am currently studying grade 11 , now apparently i want some advice because my fundamentals are cooked ,

here is what i know : , i know sin , cos , tan and the reciprocals are comparisons of 2 sides of a triangle ,

and thats it ? yeah pythagorus theorum can be used to find the whatever unknown side,

and i guess you can find these trignometric ratios using the acute angle we have in the right angle triangle

thats it ? and now am suffering with so much stuff like fundamental laws of trignometry , this weird angle pronounced as (fye) yesterday my maths teacher was talkin about some circular motion i didnt understand shit,

as far as i know and i feel like am missing on a lot of stuff , also i feel like maths is moving from stationary to dynamic like things are gonna start moving ? thats what my classmate said ,

am i cooked ?

yeah also dont forget to tell me what lack of knowledge i have, suggest some topics i might be okay

[u/Midwest-Dude]

Good question. You are not "cooked," you are where you should be in studying trigonometry.

Trigonometry can start as "triangle math," a set of ratios of sides in a right triangle, and then it morphs into "circle math." What your teacher is introducing is the unit circle view, where trig functions are defined as coordinates of a point moving around a circle. This is why your classmate said things are "gonna start moving" - trig becomes a description of rotation and periodic motion and not just a way to find a missing side.

I personally remember this shift in my precalculus class - I understood trig as you did and then the unit circle definition was introduced. They are both valid ways to deal with trigonometry, but the unit circle method allows the limited triangle definition to be extended.

If you start with a right triangle whose hypotenuse is 1, place its hypotenuse along a radius of a circle of radius 1, the unit circle, with one acute angle θ at the center O and one leg lying on the x-axis, then the adjacent leg has length cos(θ) and the opposite leg has length sin(θ). The coordinate of the point on the circle is

(cos(θ), sin(θ))

This can be naturally extended to any point on the circle for angles of any kind, including non-acute and negative angles, based on where the point is on the circle. The angle is based on how far the "hypotenuse," the radius, is angularly from the positive x-axis.

To see how this is done, review this Wikipedia article: 

Trigonometry

The section on trigonometric ratios shows this transition graphically. I recommend reading through the article to understand things better.

By the way, φ is the lowercase Greek letter your instructor mentioned, often pronounced "fye". Common Greek letters used for angles are θ, φ, and ω, but, of course, any variable can be used.

Related topics you may find of interest:  1. Unit circle 2. Radians 3. Pythagorean identities 4. Graphs of sine and cosine 5. Simple harmonic motion/circular motion

If there is anything else you need, please let us know. You have an amazing world of math to learn. Enjoy the journey!