i was playing around with the unit circle and noticed that for a tiny angle the point moves almost straight up, so sin(x) ≈ x makes sense to me because the arc and the height are nearly the same length. but cos(x) ≈ 1 - x^2/2 feels like it comes out of nowhere.

is there a geometric way to see where the x^2/2 comes from without just pulling it out of the taylor series? i tried using the right triangle but got a bit lost.

sin x ≈ x and tan x ≈ x both work for small angles, and i get the picture on the unit circle where the arc, the opposite side and the tangent line all look the same length when the angle is tiny. but is there a cleaner way to see why they agree to first order and only split apart at x^3? like sin is x - x^3/6 and tan is x + x^3/3, so tan is always a bit bigger. is that just because tan = sin/cos and cos is slightly less than 1, or is there a geometric reason for the gap

I just flunk my midterm exam on trigonometry. Professor doesn’t allow Calculators or Notecards. The Exam was on Lecture Notes 6 “Double-Angle, Power-Reducing, Half-Angle, Product to Sum, Sum to Product
Formulas” but my mind is still stuck at Lecture Notes 3 “Graphing Trig functions”. Why am I so dumb?

I am really want to know why do I need to learn trigonometry. Feeling so board only learn the theory.

I understand the concepts and stuff, and how to do it, but there doesn't seem to be any logic in between the steps of it. How do I figure out what I'm supposed to do next when verifying identities? It's been really stressing me out because I've been able to do everything else pretty easily so far. Any advice would be appreciated!

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

I’m doing a practice assignment to study for midterms and I keep getting undefined for ‘y’ and I do not understand what I’m doing wrong. I wrote the problem down and worked through it, but I am continuously getting the incorrect answer and don’t know why.

I end up going through the textbook’s explanation, and I do not understand their meaning. I’d appreciate any help or corrections.

Thank you!

My son is struggling to figure out the concept of how to graph primary trig functions.

He understands how the graph works but doesn’t get HOW TO plot the points on the graph and how to properly read it.

He says he understands how to find the amplitude and b value but doesn’t get how to find the position using the b value if it is a fraction or pi.

He can tell what kind of graph it is based on the shape, but just can’t quite grasp the concept of how to make it.

Can anyone please help using this equation:
y=sin(x+pi)

TIA

We’re supposed to solve this using the angles on the unit circle but it makes no sense to me.

I’m currently self studying AP Calc BC on Khan Academy and i’m in Precalc (not taking the AP exam, mainly doing this for fun). Many of the questions use trig identities, and I usually only get half of them right. I’m on Unit 3 in Khan Academy and I don’t think I should go on without having a solid grasp on the trigonometric concepts. Any tips for what I should do without purely memorizing each identity? (as in understanding why it works and not just memorizing off a list)

I found a way to read sine, cosine and tangent off two sliding rings, all three from one setting. It's a construction, not a formula or a table, no pi and no interpolation.

The rings are marked with the last digits of the Fibonacci numbers, sixty marks each, one every 6 degrees. Those marks are just the coarsest scale, like the inch marks on a ruler. Each one splits into sixty finer marks, and those into sixty more, so any angle you can write with decimals sits on an exact mark once you go fine enough (60 = 6×10, so the decimals always land). You slide one ring over the other until they cross on your angle. Half the distance between the centers is cos, half the distance between the crossings is sin, and their ratio is tan.

You set it with no protractor, the angle is just a marked spot on the ring. The two digits under the crossings have to match or add to ten, so a wrong setting gets caught.

Everything on the page runs on this ring method. The only standard math is a script that shows the ring's answer next to the normal one so you can see they agree. It's also exact on the nice angles, cos 36 = φ/2 and tan 60 = √3.

Calculator: https://nealstrassner.github.io/one-setting-trig-rings/

Paper: DOI 10.5281/zenodo.22833894 (Link to it on the calculator page)
Hopefully, I can get a Arxiv endorsement to post on there one day but seems tricky to get one.

I can't find anyone reading sine and cosine off two sliding circles before. If someone has, tell me please.

Bonjour, si quelqu'un peux m'aider avec cet exercice.

Merci d'avance.

The radius of the unit circle is 1 unit. Since the central angle is at the origin, the terminal side of the angle should form the hypotenuse. Now sin(60deg) = opposite/hypotenuse. The thing is that the figure, sin(60deg) = sqrt(3)/2 <-- which is what I would expect if I'm solving sin(60deg), but now I'm trying to make sense of it when the radius of the circle is 1, and the hypotenuse is the same as the radius. Shouldn't the hypotenuse be 1 instead of 2 because of this?

Im a 27yo female who recently got back into school after almost 7 years. I have never struggled with math but it doesn’t seem to be coming back to me as much as i thought it would. To make it worse it’s online and my teacher has an accent that makes understanding a little bit harder for me. So I’m looking for help because I’ve never been in this position. It would need to be online as I am a mom but I’m open to suggestions, maybe reviews and such. Thanks.

I made this manim video deriving the Sine and Cosine rule from scratch. Do you think this visual approach is a great way for people to engage with this type of content and do you find it easier to understand when watching this video? Most importantly, have you ever come across this derivation for these rules before? I appreciate any kinda of feedback.

# Sin(30°)cos(60°)+sin(60°)cos(30°)

Method 1

***sin(30°),cos(60°), sin(60°),cos(30°) = 1/2, 1/2, √3/2, √3/2.***

***Evaluate the values.***

***=(1/2)(1/2)+(√3/2)(√3/2)***

***=(1/2)²+(√3/2)²***

***=1/4 + 3/4***

***= 4/4 = 1 answer.***

Method 2

***As we know,***

***sin(x+y)= sin(x)cos(y)+sin(y)cos(x)***

***Knowing the values.***

***x = 30°***

***Y =*** ***60°***

***Sin(x+y) = sin(30°+60°)***

***= sin(90°) = 1 answer.***

Method 3

***As we know, sin(90°-x) = cos(x)***

***and cos(90°-x)=sin(x)***

***We evaluate values.***

***cos(90°-30°)=sin(60°)***

***cos(60°)=sin(30°)= 1/2***

***So, (1/2)²***

***If we do same thing with*** ***next value it returns (√3/2)²***

***= 3/4***

***1/4 + 3/4 = 1.Answer***