Lecture 2
Continuity of Trigonometric Functions
Using Squeeze Theorem
Math 13200
y
P
sin θ
O
θ
θ
cos θ
A
x
Today, we investigate the continuity and derivatives of trigonometric functions.
The first thing that we hope for is that by investigating the (continuity/differentiability) at one
x-value of sin θ, we can use the symmetry of the circle (via trigonometric identities) to find its
behavior at all points.
Continuity of sin at 0
So we first show that f is continuous at θ = 0. What does this require us to do? We must show
that
lim sin θ = sin 0 = 0
θ→0
Philosophically, this means when θ is very close to 0, the y value of Pθ is near 0.
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First step: for all θ, we have
−θ ≤ sin θ ≤ θ
This follows from the geometric picture:
The length of the line segment BPθ , corresponds to sin θ (by definition). The length of the arc
APθ (by definition of Pθ ) is θ. Since BPθ is shorter than APθ , we see that sin θ ≤ θ. A similar
computation works (on the other side of the x-axis) to show −θ ≤ sin θ.
We conclude that we indeed have
−θ ≤ sin θ ≤ θ
By letting θ be very near 0, we see that sin θ must also be close to 0 - its “squeezed” by θ (which
has limit 0 at 0) and −θ (which also has limit 0 at 0).
Squeeze Theorem Let f , g, and h be functions satisfying f (x) ≤ g(x) ≤ h(x) for all x except
possibly at c.
lim f (x) = lim h(x) = L
x→c
x→c
then we also have
lim g(x) = L
x→c
We say that g is squeezed by f and h at c.
Thus, the squeeze theorem gives us what we want! Indeed, sin θ is squeezed by θ and −θ at 0. That
is, we have
lim θ = lim −θ = 0
θ→0
θ→0
So that
lim sin θ = 0
θ→0
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A very similar argument will show that
lim cos θ = 1
θ→0
Continuity of Trig Functions
Now we can use these specific limits to establish the continuity of trig functions:
Trig Functions are Continuous The trig functions (sin, cos, tan, cot, sec, csc) are continuous
when they are defined.
Proof. We will show this for cos. Setting α = θ − θ0 , we see that θ → θ0 and α → 0 are equivalent.
We have
lim cos θ = lim cos(α + θ0 )
θ→θ0
α→0
= lim cos α cos θ0 − sin α sin θ0
α→0
= lim cos α cos θ0 − lim sin α sin θ0
α→0
α→0
= cos θ0 · lim cos α − sin θ0 lim sin α
α→0
α→0
= cos θ0
sin is done similarly, and the other 4 are done by using the fact that quotients of continuous functions
are continuous when they are defined.
Now that we have successfully investigated the continuity of trig functions, how about their derivatives? We will answer this question next lecture.
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