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4.1_Angles_and_Their_Measure

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					       Chapter 4
Trigonometric Functions
               Background

•The term trigonometry is derived from ancient
Greek roots meaning “three side measure”.
•It is the study of ratios within right triangles.
•Trigonometry came about in the ancient world
when the transition was made from a flat earth
to a world of circles and spheres.
•Today it is used to understand everything from
electrical current to modern
telecommunications.
•Any type of periodic (recurring) behavior can
be modeled using trigonometric functions.
  Section 4.1
Angles and Their
    Measure
Warm-up
Warm-up
Angle


Vertex
       90°
              Standard
        y




               Position



180°                          0°
       O Initial side
                          x




            270°
               Degree

•A degree, represented by the symbol °,
is a unit of angular measure equal to
1/180th of a straight angle.
•In the DMS(degree-minute-second)
system of angular measure, each degree is
subdivided into 60 minutes (denoted ´)
and each minute is subdivided into 60
seconds (denoted´´)
 Example 1: Working with DMS Measure


1. Convert 37.425° to DMS.




2. Convert 42°24´36´´ to degrees.
                Radians

A radian is the measure of a central angle
θ that intercepts an arc “s” equal in
length to the radius “r” of the circle.
                                     s
                                θ ,
          y




            r       s=r              r
                              where θ is
            θ
              r               measured in
                              radians.
                    x
Radians are the angle measures used in
calculus.
360° = 2π radians
       2π             360
 1                         1 radian
      360             2π
        π                         180
  1                  1 radian 
       180                         π
Formula to go from    Formula to go from
degrees to radians.   radians to degrees.
            π
        y




              rad
            2



π rad                x   0 rad


            3π
               rad
             2
   Example 2: Working with
       Radian Measure
1. How many radians are in 60 degrees?




2. How many radians are in 150 degrees?
   Example 2: Working with
       Radian Measure
3. How many degrees are in π/4 radians?




4. How many degrees are in 3π/5 radians?
              Arc Length



For a circle of radius “r”, a central angle θ
intercepts an arc of length “s” given by
                   s = rθ
where θ is measured in radians.
Example 3: Finding arc length


1. A circle has a radius of 27 inches.
Find the length of the arc
intercepted by a central angle of
160°. Give the answer in terms of π
and rounded to the nearest
hundredth.
1.   Change 160° into radians.
            π    8π
     160          rad
           180    9

2.   Find the arc length.

   8π
s     27  24π inches  75.40inches
    9
 Example 3: Finding arc length


2. Find the perimeter of a 30° slice of a
large 8in. radius pizza
Example 3: Finding arc length


3. A 100-degree arc of a circle has a
length of 7 cm. To the nearest
centimeter, what is the radius of the
circle?
 Example 3: Finding arc length


4. The concentric circles on an archery
target are 6 inches apart. The inner
circle (red) has perimeter of 37.7 inches.
What is the perimeter of the next
largest (yellow) circle?
Note: radius = 1 radian
              Example 4


John’s truck has wheels 36 inches in
diameter. If the wheels are rotating at
630 rpm (revolutions per minute), find the
truck’s speed in miles per hour.

				
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