The Unit Circle

oh okay

but their values arent determined by their distance frmo the origin, the angle determines the value, which is the distance from the origin

also,
>I.e. Quadrant 1 x values are from biggest (farthest distance away from origin) to smallest (least distance)
only works for sine, not cosine

For y values you would consider their distance from when y=0, just as determined our x-values from their distance from x=0.

I never said how we determined the three values we are going to assign, only that we can assign them systematically.

The unit circle is an easy way to get the exact values for trig functions at certain angles. Want to know the sine of pi over three? Just look at it's y coordinate on the unit circle and the answer is root three over two.

However, there is no way to get around memorizing the unit circle. You should already have the first quadrant of the unit circle memorized and if not it shouldn't be to difficult. If it is too difficult I would suggest dropping out of high school or becoming a women's studies major. Otherwise just memorize the coefficients in the numerator of the other angles, 2, 3, 5, 7, 5, 4, 5, 7, and 11. The denominators of the angles and the rest of the coordinates can all be extrapolated from the first quadrant, it's a pretty simple pattern.

is there a way to understand it in such a way where i can remember all the angles and values a year from now?

youtu.be/qTbDQ9gkKJg

Just know the unit triangles, and the shapes of the sin and cos functions.

SOH CAH TOA (sine cosine, tan, ratios) and the CAST rule (signs for each function in each quadrant).

draw 30-60-90 triangle or 45-45-90 triangle and remember the ratios. after you use it enough times you just remember

my profs would always try to draw some chart of special angle values but that shit never did me any good, grade 10 math teacher's method always stuck

I never understand exactly what people mean about this, but let me make a suggestion and a small blogpost.

When I was 14-15 and I was being taught basic high-school level geometry for the first time, and I had a vague idea of where the discussion was going, I can clearly remember appreciating that there are two special types of right triangles: the right isosceles triangle (the unique instance of such, up to similiarity) and the 30-60-90 triangle, which not only is a perfect half of an equilateral triangle, to use a vague phrase, but furthermore has its angles in elementary arithmetic progression, to use an unnecessarily complex yet appropriate one.

Sure enough, a few weeks later, the teacher started emphasizing these two triangles, which properly set us up to learn trig. And I had honestly considered them before the teacher started emphasizing them. I remember that very well.

So my real answer to the OP is: focus on those two particular right triangles. After all, everything in your OP image is obviously directly related to exactly one of those two triangles, the signs and such being simple variations on the theme. That's my real and honest answer for your question. Once you have that in mind and how the triangles are supposed to "trace" about the circle, you can draw that nice picture from memory and at will, on command.

t. math grad

You have to memorize the angles.

Pi/6 is 30 degree
Pi/3 is 60 degree
Pi/4 is 45
Pi/2 is 90

From there you can do algebra to find angles for the rest.

For the values of trig, know that the hypotenuse is always 1. Then use special triangle rules to figure out the lengths of the other 2 legs.

From there, just use the trig relations to get your answer.

its bs

youtube.com/watch?v=3GgO7Q_kg8Q&t=835s

Just memorize it you brainlet, things do not get easier than this, I promise

Trigonometric formulae.
Deriving the more sophisticated ones each time is a waste of time, yet memorizing them is hard as hell.

[math] \displaystyle
\begin{matrix}
angle & sin & cos \\
0 & \sqrt{0}/2 & \sqrt{4}/2 \\
\pi/6 & \sqrt{1}/2 & \sqrt{3}/2 \\
\pi/4 & \sqrt{2}/2 & \sqrt{2}/2 \\
\pi/3 & \sqrt{3}/2 & \sqrt{1}/2 \\
\pi/2 & \sqrt{4}/2 & \sqrt{0}/2
\end{matrix}
[/math]

>able to do higher maths
>still can't work with geometry and unit circle

life is suffering

Just convert everything into [math]e^{i\theta}[/math] form and don't convert back until you're done.
They're easier to manipulate than trig functions, and any geometric intuition you'd be using in trigonometry carries over to the complex vector interpretation.

>">able to do higher maths"
You are just fooling yourself. If you can't solve IMO math problems then you can't do higher maths. This is a HUGE problem in the science fields. People want to learn "higher" or "advanced" techniques without mastering what they consider to be "basic", just because they think its more "cool" or it has more fancy symbols. It is way more valuable to be a fucking master in polynomials than knowing all the formulas on an analysis book, being an expert at the "basics" can take you far, and people don't seem to believe that.

t. buttmad engineer who failed intro topology

By far the best

Tried to watch, but fapped instead.

Do you have a video of some poo in loo explaining it?

>chart not simplified

Into the trash. My professors would laugh and fail me on a test if I wrote "sqrt4/2" in a rigorous class like calculus 2.

Not being a retard.

jej'd

PatrickJMT is literally a god

youtube.com/watch?v=3GgO7Q_kg8Q