Transcript Activity 7: Investigating Compound Angles

Activity 7: Investigating
Compound Angles
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D
A
B
C
Activity 7: Investigating Compound Angles
•In the introduction of the activity, we conclude that the best way to
find the exact value of sin75o was to write is as a sum of two
angles:
sin(75)
 sin(30  45)
•We know that:
  
 sin  
6 4
  1
sin  
6 2
2
 
sin  
2
4
•So, if:
  
sin   then,
6 4
2 1 2
   1
sin    

 1.207
2
2
6 4 2
Activity 7: Investigating Compound Angles
•When we try to get an approximate value on the calculator we get:
  
sin  
6 4
 0.966
•Why do we get two different answers?
•We have to graph this angle in standard position to verify which
answer is correct.
Activity 7: Investigating Compound Angles
•Let us place this angle in standard form where radius is 1:
y
1
β
π/6
π/4o
75
sin(π/6)cos(π/4) sin(π/4)cos(π/6)
HYPOTENUSE
π/4
OPPOSITE
Therefore,
We
use
the
cosine
ratio:
Given
the
angles
inright
the
We
can
solve
for
this
right
are
now
going
to
create
Since
The
altitude
we
are
of
trying
the
to
find
Split
Using
the
simple
angle
geometry
into
radian
and
To
solve
for
the
second
o)=sin(π/6+
sin(75
π/4)
smallest
triangle,
itour
isπ/6
triangle
we
have
aaltitude
right
triangle
inside
the
ratio
created
sin(π/6+
with
π/4),
the
we
angles
creating
ofsince
aright
π/6
pair
+
of
π/4.
parallel
This
is
we
must
find
the
cos(π/4)=AD/HY
o calculate
othe
now
possible
to
the
angle,
π/6
and
compound
angle
where
should
angle
π/4
identify
can
be
the
solved
sides
the
lines
same
we
see
as
45
that
and
the
30
blue
.we
dot
denoted
by
the
blue
sin(π/6)cos(π/4)+sin(π/4)cos(π/6)
cos(π/4)=Altitude/sin(π/6)
altitude.
hypotenuse
of
right
angle
on
the
end
need
since
in
itsorder
hypotenuse
to1.this
solve
is
this
The
and
the
red
isstart
dot
now
must
ausing
sum
up
dot
Whenangle
we
calculate
special
of
the
green
arrow.
ratio.
cos(π/6).
compound
to
90o.sin(angle)=OP/HY.
Based
angle.
on alternate
angles:
Altitude=sin(π/6)cos(π/4)
 2 blue


x angles,
 1  2  the
 3  dot must be

 
  2  2 
 2  2 to
 π/4
 rads.

equal
=0.966
This is the answer we got on
the calculator
Activity 7: Investigating Compound Angles
•In general, when finding the sine ratio of a compound angle as
shown below:
sin(A+B) = sinAcosB + sinBcosA
B
sinAcosB+sinBcosA
A
Activity 7: Investigating Compound Angles
•Go back to the activity website and complete the rest of the activity
online!