exact value of sin(3 degrees)

In this video, we will find the exact value of sin(3 degrees). We will see the special special triangles and the angle difference formula! ft. sin(15), sin(18), cos(15), cos(18)

[stewartzayat]

Mathematician:
Physicist: sin(3°)=pi/60
Engineer: sin(3°)=0

[GreenMeansGOF]

11:03 *thought download begins*
12:07 *thought download complete*

[hamiltonianpathondodecahed]

this question is very easy using the fundamental theorem of engineering

*sin x ≈ x* | x in radians |
*π ≈ 3*
using these we get the
answer as 0.05
%error of 4.5%

[DjVortex-w]

At 11:05 you can almost hear the cogwheels turning in his head...

[alexharkler]

3:03 "Of course, 1 is equal to 2"
-BPRP 2019

[bruhmoment]

Bprp: *Runs math channel like a boss*

Also bprp: *1=2*

[kennylim]

Really shows you even an expert has troubled moments

[jeremyzamayla]

"1+1+1 =3"
We did it boys, an A in maths
👏👏👏

[IoT_]

I work as a teacher of control systems which involves a lot of different math subjects. Thank you for showing HOW TO TEACH STUDENTS. I like how you tell in detail mathematics. I really appreciate it.

[Periiapsis]

For sin and cos of 15° couldn't you have also used the difference formula for sin and cosine?
sin(45 - 30) = sin(45)cos(30) - cos(45)sin(30)
cos(45 - 30) = cos(45)cos(30) + sin(45)sin(30)

[spinningcycloid]

He teaches very friendly. Even for a simple calculation, he explains very kindly. So I can understand whole topic. Thank you for your works!

[Mihau_desu]

"1 is equal to 2"
- bprp 2019

Btw. Great video

[Metalhammer]

Wohoo I'm not the only one deriving the angle sum from Euler's formula! My professor thought me mental xD. Didn't subtract points but asked me if I'm slightly troubled that I find that simpler than geometric proofs xD

[patrickmckinley]

Digging through some of my old papers, I found where I ran this calculation years ago . I just ran the half angle formula on 30 degrees to get the sin and cos of 15 degrees. I love your construction to do it geometrically - never seen that before.

[SyberMath]

Nice problem! I have 2 comments:
1. 23:52 it's easier to just say "divide the hypotenuse by sqrt(2) to get the leg so it's sqrt(3)/sqrt(2)"
2. 24:44 the second leg should be sqrt(3)/sqrt(2) not sqrt(3)/sqrt(3) 😊

[whyit]

I like how this went back to your old video about special phi triangles! Also, I loved how there's such an elegant way to find an exact sine of an angle! Great job on the video.

[sharmsma]

The result should be almost equal to pi/60. For small angles, sin x approximates x with x in radians. Converting 3 degrees to radians is just multiply with pi/180.

[hussiensayed]

Can we do this without triangels
1]for 18°
For this equation sin(X)=cos(4X)
X=18° satisfies the eq where 4*18=72=90-18
We know cos(4X)=2cos^2(2X)-1
=2(1-sin^2(x))^2-1
Let y=sinx
Then y=1+8y^4-8y^2
8y^4-8y^2-y+1=0
This eq has 4 solutions but one of them is sin18
8y^2(y^2-1)-(y-1)=0
(y-1)(8y^2(y+1)-1)=0
y=1 is a sol but not sin 18 cuz sin90=1

8y^3+8y^2-1=0
8y^3+4y^2+4y^2-1=0
4y^2(2y+1) + (2y-1)(2y+1)=0
(2y+1)(4y^2+2y-1)=0
y=-0.5 is a sol but not sin18 cuz sin 210=-0.5

4y^2+2y-1=0
y=(-2±sqrt(16+4)) /(2*4)
=0.25(-1±sqrt(5))
Two solutions but we have one +ve solution and we know sin 18 is +ve

Then sin 18° =0.25(-1+sqrt(5))
Cos 18° =sqrt(1-sin^2 (18))
=sqrt(1-(6-2sqrt(5))/16)
=sqrt(5-sqrt(5))/2sqrt(2)

2]for 15°
Cos 30=2 cos^2(15)-1
Cos15=sqrt((1+sqrt(3)/2)/2)
=sqrt(4+2sqrt3)/2sqrt2
=sqrt(3+2sqrt3+1)/2sqrt2

=sqrt( (sqrt3+1)^2 ) /2sqrt2
=(sqrt3+1)/2sqrt2

Sin15=sqrt(1-cos^2(15))
=sqrt((1-sqrt(3)/2)/2)
=sqrt(4-2sqrt3)/2sqrt2
=sqrt(3-2sqrt3+1)/2sqrt2

=sqrt( (sqrt3-1)^2 ) /2sqrt2
=(sqrt3-1)/2sqrt2

3] finally sin 3°=sin (18°-15°)
=sin18°cos15°-cos18°sin 15°

[jensonjoseph]

I learnt a lot of special specials for the 1st time, though I knew sin (18) and sin (15) algebraically. Also the proofs of sin (a-b). Thank you, you are a special special teacher : )

[bayanmehr]

Super fun video :) I love how you talk about angles like they are people