is pieequal to 360 or is it equal to 180??

Question

1. pie=360 0r 120???
2. proof (cos x)'= -sin x

Answer 1

Question 1
2π = 360°
π = 180°

Quesstion 2
f (x) = cos x
f `(x) = - sin x
Proof
f `(x) is given by:-
lim h->0 [ cos(x + h) - cos x ] / h
lim h->0 [cos x cos h - sin x sin h - cos x] / h
lim h->0[(cos x cos h - cos x)/h - sin x(sin h)/h ]
= - sin x

Answer 2

Basic relation:
Length of circumference = 2*pi*r (where r is the radius).

If you apply the length r on the arc, you find that you need to apply it (2*pi) times in order to get all the way around. Otherwise, the basic relation would be false.

The angle needed at the centre of the circle to subtend the arc of one radius is the same, whatever the radius of the circle. This is the angle we call one radian (the name is formed from the word radius).

There are nice things about using radians. For example, the trigonometric functions can be replaced with series:

for example:

cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...
and
sin(x) = x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...
where ! indicates the factorial of the number
(e.g., 4! = 1*2*3*4 = 24)

(I use the D notation for differentiation, easier on this screen)
(cos x)' =
D[cos(x)] =
D[1 - (x^2)/2! + (x^4)/4! - (x^6)/6! + ...] =
D[1] - D[(x^2)/2!] + D[(x^4)/4!] - D[(x^6)/6!] + ...

one by one:
D[1] = 0
D[(x^2)/2!] = 2(x^1)/2! = (x^1)/1! = x
D[(x^4)/4!] = 4(x^3)/4! = (x^3)/3!
D[(x^6)/6!] = 6(x^5)/6! = (x^5)/5!
...
hint: 6 / 6! = 6 / (1*2*3*4*5*6) = 1/(1*2*3*4*5) = 1/5!

Putting it all together:
D[cos(x)] = 0 - x + (x^3)/3! - (x^5/5!) + ...
D[cos(x)] = - [ x - (x^3)/3! + (x^5)/5! - (x^7)/7! + ...]
D[cos(x)] = - [sin(x)]


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Answer 3

pi (in radian) is equal to 180 (in degrees) for the measurement of an angle

Answer 4

pie radian is 180 degree

Answer 5

Pie is not equal to either of those. Pie is 3.14
There are 360 degrees in a circle though.

Answer 6

There are 2*pi radians in a circle.

Answer 7

pie is equal to 3.14.... if we're talking about the same thing. a triangle is 180, a circle is 360, but pie is 3.14

Answer 8

you mean pi radian

pi radian = 180 degrees

for 2

d/dx(cos x) = -sin x

Proof:

(1) Let y = cos(x)

(2) dy/dx = lim(Δx → 0) [ cos(x + Δ x) - cos(x) ]/Δx

(3) Now

cos(x +Δx) - cos(x) = -2*sin[(x + Δx + x)/2]*sin[(x+Δx-x)/2] =
= -2*sin(x + Δx/2)*sin(Δx/2)

(4) Once again, applying the Product Rule for limits gives us:

dy/dx = lim(Δx → 0) [ cos(x + Δ x) - cos(x) ]/Δx =
lim(Δx → 0)[ -sin(x + Δx/2) ] * lim(Δx → 0) [ sin (Δx/2)/(Δx/2)]

(5) Again, setting θ = Δx/2 gives us:
lim(Δx → 0)[ sin(θ)/θ ] = 1
(6) also :
lim(Δx → 0)[ -sin(x + Δ x/2) ] = -sin(x)

(7) Putting this all together gives us:
dy/dx = -sin(x)*1 = -sin(x)

Answer 9

pi (rad) = 180 (deg)

prove that (cos x)' = - sin x....

http://math2.org/math/derivatives/more/trig.htm