AP Calculus / Mrs. DeMann Day 2
Limits involving Trig Functions
If , what is the value of x?
Squeeze Theorem: If for all in an open interval containing , and if
, then
Two special trig limits: , and
Proof: We will use a unit circle (radius = 1, center (0,0)), and for a variable instead of x, so as not to confuse with the x-axis. Our functions that we will use to “squeeze” will be area functions of 2 triangles and the area of a circle sector.
Given a unit circle, with a central angle where .
Brief Review:
1. What is the formula for the length of an arc?
2. What is the area of a circle sector?
3. Useful Trig Identitites:
Proof:
We will use these two special trig limits frequently, so it is helpful to understand their limits.
I) II)
We will use the area of two triangles and a circle sector (pie shape), “squeezing” the circle sector between the area of the two triangles.
On the circle diagram, label the following points:
1) circle center: O
2) (1, 0): B
3) (): P
4) (): Q
Consider the area of : Height of =
area =.
Now consider : Height of =
Area =
What is the area of the circle sector?
Given , write an inequality for the areas.
Now let’s do some algebra. Multiply everything by .
The result is:
. Take the reciprocals, remembering to switch the inequalities:
. Now we can take the limit of the outside expressions:
, and . Since both limits = 1, and
is always in between these two functions, then .
Now we need to find . Again, direct substitution does not work. Let’s consider a comparison between the length of and . From our knowledge of geometry, we know that given any two points on a circle, the segment will always be less than the arc length. Using the distance formula , cleaning this up:
, so . . Setting up an inequality:
. Square everything, and multiply by . The result is:
. Now take the limit of the outside functions.
, and , so, since will always be in between, then .
Examples:
1)
2)
HW: Exercises P. 68 #67 - 78