In today's blog, I will show how it is possible to use a unit circle to establish that if x is greater than 0 and less than π/2, then sin x is less than x which is less than tan x.
Theorem: if x is in (0,π/2), then sin x is less than x is less than tan x
Proof:
(1) Let O be a circle with radius = 1.
(2) Let ∠ CDO and ∠ BAO be right angles.
(3) Let x be the angle at ∠ COD
(4) We can see the following values apply to this diagram
cos x = adjacent/hypotenuse = OD/OC = OD/1 = OD
sin x = opposite/hypotenuse = CD/OC = CD/1 = CD
tan x = sin x/cos x = opposite/adjacent = AB/OA =AB/1 = AB
(5) Area of triangle OAC = (1/2)(base)(height) = (1/2)(OA)(CD) = (1/2)(1)(sin x) = (1/2)(sin x) [See Lemma 2, here for proof]
(6) Area of sector OAC = (1/2)(x)(radius)2 = (1/2)(x)(1)2 = (1/2)x [See Lemma 2, here for proof]
(7) Area of triangle OAB = (1/2)(base)(height) = (1/2)(OA)(AB) = (1/2)(1)(tan x) = (1/2)(tan x) [See Lemma 2, here for proof]
(8) By the diagram above, it is clear that triangle OAB is greater than sector OAC which is greater than triangle OAC.
(9) This then gives us that (1/2)(tan x) is greater than (1/2)(x) which is greater than (1/2)(sin x).
(10) Dividing all values by (1/2) gives us:
tan x is greater than x which is greater than sin x.
QED
References
Thanks for the lesson. It helped me understand the squeezing theorem in my Calculus book.
ReplyDeleteUNC Greensboro Student
Thanks for your effort.
ReplyDeleteThis is what I had tried before landing at your blog.
Area of triangle OCD is less than the sector's area
=> (1/2)*sin x *cos x <(1/2)*x
And then I was stuck...
I thought that the inequality could be written as
sin(2x) < (2x), 0< (2x) < (pi/2)
Assuming y = 2x,
sin y < y, 0< y < (pi/2)
Is this a valid proof? Or am I kidding myself?
Thanks for your time
Very beautiful proof! Thank you very much for posting it and explaining it with so many details!!!
ReplyDelete