Transcript Angles, Degrees, and Special Triangles

Radians and Degrees
Trigonometry
MATH 103
S. Rook
Overview
• Section 3.2 in the textbook:
– The radian
– Converting between degrees and radians
– Radians and Trigonometric functions
– Approximating with a calculator
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The Radian
Motivation for Introducing Radians
• Thus far we have worked exclusively with angles
measured in degrees
• In some calculations, we require theta to be a real
number – we need a unit other than degrees
– This unit is known as the radian
• Many calculations tend to become easier to perform
when θ is in radians
– Further, some calculations can be performed or even
simplified ONLY if θ is in radians
– However, degrees are still in use in many applications
so a knowledge of both degrees and radians is
ESSENTIAL
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Radians
• Radian Measure: A circle with
central angle θ and radius r
which cuts off an arc of length
s has a central angle measure of
s
 
where θ is in radians
r
– Informally: How many radii r comprise the arc length
s
• For θ = 1 radian,
s=r
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Radians (Example)
Ex 1: Find the radian measure of θ, if θ is a
central angle in a circle of radius r, and θ cuts
off an arc of length s:
r = 10 inches, s = 5 inches
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Converting Between Degrees and
Radians
Relationship Between Degrees and
Radians
• Given a circle with radius r, what arc length s is
required to make one complete revolution?
– Recall that the circumference measures the distance
or length around a circle
– What is the circumference of a circle with radius r?
C = 2πr
• Thus, s = 2πr is the arc length of one revolution and
is the number of radians in one
s 2r
  
 2
revolution
r
r
• Therefore, θ = 360° = 2π consists of a complete
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revolution around a circle
Relationship Between Degrees and
Radians (Continued)
• Equivalently: 180° = π radians
– You MUST memorize this conversion!!!
• Technically, when measured in radians, θ is
unitless, but we sometimes append “radians”
to it to differentiate radians from degrees
– Like radians, real numbers are unitless as well
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Converting from Degrees to
Radians
• To convert from degrees to radians:
– Multiply by the conversion ratio  rad
so that degrees will divide out 180 
leaving radians
– If an exact answer is desired, leave π in the final
answer
– If an approximate answer is desired, use a
calculator to estimate π
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Converting from Degrees to
Radians (Example)
Ex 2: For each angle θ, i) draw θ in standard
position, ii) convert θ to radian measure using
exact values, iii) name the reference angle in
both degrees and radians:
a) -120°
b) 390°
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Converting from Radians to
Degrees
• To convert from radians to degrees:
– Multiply by the conversion ratio 180 
so that radians will divide out  rad
leaving degrees
• The concept of reference angles still applies
when θ is in radians:
– Instead of adding/subtracting 180° or 360°,
add/subtract π or 2π respectively
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Converting from Radians to
Degrees (Example)
Ex 3: For each angle θ, i) draw θ in standard
position, ii) convert θ to degree measure, iii)
name the reference angle in both radians and
degrees:
a)
11
6
b)
 5
4
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Radians and Trigonometric
Functions
Radians and Trigonometric
Functions
• On the next slide is a table of values you
should have already memorized when θ was
in degrees
• Only difference is the equivalent radian values
– Each radian value can be obtained via the
conversion procedure previously discussed
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Radians and Trigonometric
Functions (Continued)
Degrees
Radians
cos θ
sin θ
0°
0
1
0
30°
45°

6

4
3
2
1
2
1
2

2
2
1
2
1
2

2
2
3
2
60°

3
90°

2
0
1
180°

-1
0
270°
3
2
0
-1
360°
2
1
0
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Radians and Trigonometric
Functions (Example)
Ex 4: Give the exact value:
5
3
a)
cos
b)
7
csc
6
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Radians and Trigonometric
Functions (Example)
Ex 5: Find the value of y that corresponds to
each value of x and then write each result as
an ordered pair (x, y):
y = cos x
for
x  0,
  3
,
4 2
,
4
,
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Approximating with a Calculator
Approximating with a Calculator
• When approximating the value of a
trigonometric function in radians:
– Ensure that the calculator is set to radian mode
• ESSENTIAL to know when to use degree mode
and when to use radian mode:
– Angle measurements in degrees are post-fixed
with the degree symbol (°)
– Angle measurements in radians are sometimes
given the post-fix unit rad but more commonly
are given with no units at all
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Approximating with a Calculator
(Example)
Ex 6: Use a calculator to approximate:
a) sin 1
b) cos 3π
2 

c) cot  
 7 
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Summary
• After studying these slides, you should be able to:
– Calculate the radian measure of a circle with radius r
cutting off an arc length of s
– Convert between degrees & radians and vice versa
– Apply previous concepts such as reference angles to angles
measured in radians
– Use a calculator to approximate the value of a
trigonometric function in both degrees and radians
• Additional Practice
– See the list of suggested problems for 3.2
• Next lesson
– Definition III: Circular Functions (Section 3.3)
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