Download 7.1 Measurement of Angles

7.1 Measurement of Angles
-If you leave this class with anything this
year, it will be a better understanding of
sine/cosine/tangent and angle
measurments
• In trigonometry an angle often represents a
rotation about a point.

" theta "
Initial Ray
A common unit for measuring angles is the revolution, a complete circular motion.
The unit for measuring smaller angles is degrees, of which there are 360 degrees in
one complete revolution.
• There are 2 ways that degrees can be expressed
they can be written in decimal form or they can
be written with degrees/minutes/seconds
When measuring an angle and it falls between two whole number
degrees and we simply do not want to round that value we can be more
precise with our measurement and break 1 degree down into 60 minutes,
and then break 1 minute down into 60 seconds. Here is the notation.
o
74 32 ' 45''
Often we like to convert between these 2
different forms.
• To convert decimal degrees into
degree/minute/seconds
– Subtract away your whole number degree
• Multiply remaining decimal by 60, then take the
resulting value, subtract away the whole number and
that whole number is your minute value.
• With the resulting decimal from the previous step
multiply that decimal by 60 and the resulting whole
number is your seconds value.
– Round your seconds for this class.
39.452
o
Going from minutes/seconds to decimal
• Take your degree value and put that to the left
of your decimal.
• Take your minute part and divide by 60.
• Take your second part and divide by 3600
• Sum the previous three decimals and you now
have a decimal degree equivalent to the
minute/second format
o
39 27 '7 ''
• We know of one unit of measuring angles and that is
through the use of degrees. A second unit is called
RADIANS. (remember the mode button on your ti83/84s?)
• Here is a comparison, we can measure the distance
between two objects in terms of feet or we could
measure the distance in terms of meters. Both provide
valid measurements but the values are completely
different.
• Degrees and Radians are the same way, they both
measure the same thing (an angle) but the values are
completely different yet both valid. Radians seem
difficult to use initially because they are relatively new
to us they are something we have never dealt with
before
• When an arc of a circle has the same length as
the radius of the circle the measure of the
central angle is then 1 radian.
Basically we want
to think of how
many radiuses can
fit into the arc, and
that tells you how
many radians the
central angle is.
• Radian measure of the central angle is the number or
radius units in the length of the intercepted arc. Hence
the name RADIAN
• So now if the central arc is theta it can be measured by
taking the arc length and dividing it by the radius
length.
s arc length

r
radius
• If we think about the entire circle and want to know the
measurement of the entire circle in terms of radians, we look at the
circumference. We know the C=2πr thus the arc of the entire circle
would be 2πr. We also know that the radius of any circle can be
shown as r.
s 2 r
so now   
 2
r
r
• So one entire revolution around the circle would be
2π radians. So now we can make the argument that
360⁰=2π radians.
180⁰= π radians
This allows for the 2 following conversion formulas:
1 radian 
1deg ree 
180o


180
o
deg rees
radians
We can now convert radians into
degrees and degrees into radians
(much like we convert other measurements)
• Helpful hint, in converting these 2 units you always want the initial
units to disappear, thus the conversion rate that you multiply by
must have that unit in the denominator. The only 2 numbers we use
in our conversions are 180⁰ and π.
• CONVERT THE FOLLOWING
o
325
152
o
3.2 radians
2
radians
3
Angle measurements that can be expressed in
degrees as whole numbers cannot be expressed
in radians as whole numbers and vice versa.
Therefore most of the time radians are given as
fractional parts of π. The following 3 slides
provide the angle measurements that we use
most frequently they will be presented in
degrees and radians.
It is essential that you memorize
these, you will be quizzed on them
frequently
Now all together as one
Tomorrow I will
teach a quick and
easy way to
remember how to
reproduce this
picture.
• When an angle is shown in the Cartesian plane it is
usually shown in standard position. Which means that
the angles vertex is at the origin and its initial ray is
along the positive x axis.
• The terminal ray may move counterclockwise (initially
up) and the angle is referred to as being positive.
• If the terminal ray moves clockwise (initially down)
then the central angle is referred to as being negative.
• Show examples on the board
• Sometimes we can place two angles on the same
coordinate plane. If both angles are in standard
position and they have the same terminal angle, then
the angles are said to be COTERMINAL. For example
and angle of 20⁰ and 380⁰ would have the same
terminal ray, therefore they would be coterminal.
• An angle of π/4 and 9π/4 would be coterminal
• Identifying coterminal angles in degrees
– Initial angle + k360⁰ (where k is the number of
revolutions)
• Identifying coterminal angles in radians.
– Initial angle + k2π
• Find 2 coterminal angles of
degrees
2 in radians and

3
Homework pg. 261 3-8, 11, 13, 14, 17-19