Sec 3.4 Arc Length
Arc length = the length of an arc along the outside of a circle
MTH 112 - April 20, 2009
1.
2.
Today we will . . .
- talk about arc length and area of a sector (3.4)
- hand back and discuss tests
New HW
3.4: 1, 5, 10, 11, 13, 15, 27, 31, 38, 40, 43, 53, 55
3.5: 5, 8, 11, 13, 16, 24, 29, 35 - 41(odd), 49, 51, 55
4.1: 1 - 7, 10, 11, 13 - 17, 21 - 25, 27 - 33(odd)
4.2: 1, 5, 7, 9, 12, 13, 15, 19 - 23, 25 - 33(odd), 41
4.3: 4, 5, 9, 11, 15, 17, 22, 23, 26, 29, 30, 33, 38, 41, 51
3.
Notebook labeled "MTH 112 - Trimpe" in the Learning Center
will have copies of the test keys and QUIPP keys.
4.
Remember that Tech Scholar applications are due Friday!
We used this already when defining radian measure. Remember?
θ=
where S = arc length, r = the radius, and θ = the central angle (in radians) intercepted by the arc
We can rewrite this same formula to find arc length.
S = rθ
BUT θ must be in radians!!
EXAMPLE: Section 3.4, #4
Find the length of arc S, cut off by θ, where θ = 2.4 and r = 1.8 ft.
Jan 4­5:52 PM
What if we don't have radians????
S
r
Apr 20­12:21 PM
EXAMPLE: Section 3.4 #27
A ferris wheel, known as the Great Wheel, was built in Vienna in 1897.
The diameter of this wheel is 197 feet. Find the distance traveled by a rider in going from initial position Po to position P1 if θ has the following measure.
Convert the angle to radians first.
EXAMPLE: Suppose θ = 112o, r = 8.1 cm
What is the arc length?
θ = 60o
θ
P1
Po
Change θ to radians first.
θ = 112o x
π
180o
S = rθ = 8.1 cm x =
112π
180
112π
180
(the degrees cancel)
= 15.83 cm ≈ 16 cm (only 2 significant digits)
Apr 20­12:22 PM
Apr 20­12:23 PM
1
Area of a sector
Areasector = 1 r 2θ
2
but only when θ is in radians!!
A sector is a slice of a circle ­­ like a piece of
pepperoni pizza.
θ
To find the area of a sector you need to know what part of the circle your piece is. The easiest way to do this is to use θ, the central angle of the sector.
EXAMPLE: A piece of pizza has a central angle of 36o. If the radius of the pizza is 6 inches, find the area of the piece of pizza.
Areasector = Apr 20­12:23 PM
Apr 20­12:23 PM
Specific comments on test questions
1.
Use order of operations to evaluate 5 - 2 sin 2x.
2.
Do you need a common denominator to multiply fractions?
3.
Use the quadrant to determine the sign of sine, cosine, and tangent.
Looking back over Test #1
Test Results:
Mean = 72.5
Median = 74
If you did not do as well as you expected on the test, consider:
• Did you come to class on the review day?
• Did you finish and review all your homework prior to the test?
• Did you ask for help with any problems you couldn't do?
• Did you re­do some homework problems WITHOUT looking at
your homework? • If so, did you mix up the order of the problems?
• Did you work some of the optional review problems?
• Did you make a "helpful" help card? Did you use it on the test?
• Did you learn the EXACT values for "nice" angles and memorize
the basic identities?
• Were you prepared to use any of the three different definitions of
the trig functions ­ for example,
sin θ = y/r
sin θ = opposite/hypotenuse
sin θ = y­coordinate of a point on the unit circle
• Were you well­rested when you took the test?
Apr 20­12:24 PM
Apr 20­12:25 PM
2
Specific comments (continued)
4.
Remember that a reference angle is ALWAYS:
- positive
- less than 90o
Specific comments (continued)
5.
4.
If csc θ = 2.5672, then what does sin θ equal?
5.
Simplify your answers!
How do you use a reference angle to evaluate cot(5π/6)?
How can you simplify √1 ?
What about
1
?
√10/3
Apr 20­12:29 PM
Apr 20­12:32 PM
3