I had advanced calculus in high school, along with physics, but NEVER has the sine and cosine made this much sense at any point of my education. I'm watching this 2 years after graduating law school and I'm honestly astounded at what was right in front of me and I just missed it then.
A couple videos ago, Mr. Woo said that tan is the length of the line tangent to the circle perpendicular to the hypotenuse, between where it touches the circle and where it intersects the x axis. Could someone prove this to me?
pick a point A on the unit circle point A has coordinates (cos θ, sin θ) and defines a radius OA where O is the centre (the origin) for OA: the rise is sin θ and the run is cos θ the gradient is therefore tan θ (rise ÷ run) the gradient of the PERPENDICULAR line at A is therefore -cot θ (the negative reciprocal of tan θ) notice! the perpendicular in question is, by definition, tangent to the circle at A the perpendicular has equation y = mx + c, and we know m: it is -cot θ we also know an x value and a y value on this line: (cos θ, sin θ) substitute in to find c: sin θ = -cot θ × cos θ + c so c = sin θ + cot θ × cos θ = cosec θ (you'll need to simplify the trig yourself to see why this is so) so the equation of the tangent at A is y = (-cot θ) x + (cosec θ) the tangent meets the x-axis when y = 0, which is solved when x = sec θ we therefore seek the distance from A (cos θ, sin θ) to the point on the x-axis (sec θ, 0) this is accomplished via Pythagoras' Theorem the square root of { (cos θ - sec θ)² + (sin θ - 0)² } = the square root of { sec² θ - 1 } = tan θ, as required
I'm afraid " guess" is not my bag - Reminds me of what Zack says in The Big Bang theory: " that's what I love about science, there's no one right answer. " There's no guessing in maths - there's inspiration and or intuition, and perspiration, but no guessing.
I had advanced calculus in high school, along with physics, but NEVER has the sine and cosine made this much sense at any point of my education. I'm watching this 2 years after graduating law school and I'm honestly astounded at what was right in front of me and I just missed it then.
This man is single handedly keeping me sane rn lol beyond grateful for this content.
Eddie woo is the goat. literal talking , math dictionary in my phone (O-O)
Another amazing video Eddie.
You're a star!
Btw, is that a Citizen Eco-Drive on your wrist? 😀
So the maximum and minimum in the math is equivalent to the kinetic energy of stationary wave in physics??
A couple videos ago, Mr. Woo said that tan is the length of the line tangent to the circle perpendicular to the hypotenuse, between where it touches the circle and where it intersects the x axis. Could someone prove this to me?
pick a point A on the unit circle
point A has coordinates (cos θ, sin θ) and defines a radius OA where O is the centre (the origin)
for OA: the rise is sin θ and the run is cos θ
the gradient is therefore tan θ (rise ÷ run)
the gradient of the PERPENDICULAR line at A is therefore -cot θ (the negative reciprocal of tan θ)
notice! the perpendicular in question is, by definition, tangent to the circle at A
the perpendicular has equation y = mx + c, and we know m: it is -cot θ
we also know an x value and a y value on this line: (cos θ, sin θ)
substitute in to find c: sin θ = -cot θ × cos θ + c
so c = sin θ + cot θ × cos θ = cosec θ (you'll need to simplify the trig yourself to see why this is so)
so the equation of the tangent at A is y = (-cot θ) x + (cosec θ)
the tangent meets the x-axis when y = 0, which is solved when x = sec θ
we therefore seek the distance from A (cos θ, sin θ) to the point on the x-axis (sec θ, 0)
this is accomplished via Pythagoras' Theorem
the square root of { (cos θ - sec θ)² + (sin θ - 0)² } = the square root of { sec² θ - 1 } = tan θ, as required
Rokker815 Wow
@@Rokker815 👏👏
When I look for a motivation to learn math, I look at your video
Can you tell about 3D domain function
I'm afraid " guess" is not my bag - Reminds me of what Zack says in The Big Bang theory: " that's what I love about science, there's no one right answer. " There's no guessing in maths - there's inspiration and or intuition, and perspiration, but no guessing.
Yeah, that is exactly what he said at the very end
7:31
richard feynman
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