Download Trigonometry - Miami Beach Senior High School

Trigonometry
Sine
S–O–H
Cosine
C–A–H
Tangent
T–O–A
a⁰ + b⁰ = 90⁰
a⁰ + b⁰ = 90⁰
a⁰ + b⁰ = 90⁰
Inverse Sine
Arcsin (x) or
-1
Sin (x)
-1
Sin (a/c)=α⁰
Inverse Cosine
Arccos (x) or
-1
Cos (x)
-1
Cos (b/c)=α⁰
Inverse Tangent
Arctan (x) or
-1
Tan (x)
-1
Tan (a/b)=α⁰
Solving Right Triangles
Given two sides.
1) Use the Pythagorean
Theorem to find 3rd
side.
2) Use preferred
inverse trig ratio to find
angles.
Solving Right Triangles
Given one side and one angle.
1) Use preferred trig
ratio to find 2nd side.
2) Use the Pythagorean
Theorem to find 3rd
side.
3) Use Triangle Sum
Theorem to find 2nd
angle.
Similar Right Triangles
Small
Triangle
Separate the triangles!
Medium Triangle
Large Triangle
What are the angles?
Remember, a + b = 90°
Example: Given Sides
The
triangle
isthe
similar
to2the
medium
Now
complete
large
triangle.
Thesmall
small
Which
triangle
triangle
is has
now
complete.
quantities?
Large
Medium
Small
Triangle
Triangle
Triangle
triangle.
2
22 theb
2 triangles?
What do you2know
about
=16
2other
15
+
b
=
25
9 +12 =a
Example: Given Angle & Side.
Remember,
when
you know
one
angle,
you can find
Complete
Finally,
complete
the
medium
the
large
triangle.
side.
Which
triangle
has
the
side?
the others.
10√3
10
20
The Shadow Problem
The Washington Monument casts
tree casts
a shadow…
flagpole
casts
a shadow…
aAshadow…
The
Washington
If the
flagpole is
shadow
isMonument
285’
35’ tall
long
isand
169m
tall.
If
the
shadow
the sun’s rays makes
isan
100m
long,
what
angle
angle
of 55⁰
of
40⁰
with
with
the
does
the
sun’s
rays
make
the flagpole,
tree,
how
tallhow
is the
long
tree?
is
with
monument?
the the
Round
shadow?
to
the Round
nearesttofoot.
the
th.
Round
to
the
nearest
100
th
nearest 10 .
Angles of Elevation/Depression
Which one you use depends on your
point of view. However, they are
essentially the same thing. Why?
Angles of Elevation
The Ramp
Problem
The smaller the angle, the easier
it is to push something up the
ramp. If the bed of a truck is 3.5’
high and you want a ramp with a
10⁰ angle, how long (along the
diagonal) does the ramp have to
be? Round to the 10th.
Angles of Depression
The Lighthouse Problem
A lighthouse keeper is looking
at a boat at sea. If his line of
sight is 20⁰ from the horizontal
and the light is 95’ from
sea level, how far is the
boat from shore? Round
to the nearest foot.
Angles of ????
The Ski Slope
Problem
A downhill racer skis a
mountain with a 28⁰ grade
(against the horizontal). If the
vertical height of the slope is
4,500‘ and he completed the
run in 2.5 minutes, how fast
did he go? Find the answer in
feet/minute, then mph. Round
to the nearest mile.
Angles of ????
The Kite/Balloon/Plane Problem
During the Civil War, the Union Army
used hot air balloons for
reconnaissance. If the balloonist is
1000’ high and he sees cannons
when the angle of his line of sight
is 12⁰, how far are the cannons
from his base? Round to the
nearest foot.
Which to use? Sin or Cos or Tan?
Do you have
or do you
want the
hypotenuse?
No
Use
tangent. If
you want
the angle,
use inverse
tangent.
Yes
Which do you
have/want –
opp or adj?
Opp
Adj
Use
Cosine. If
you want
the angle
use inverse
cosine.
Use Sine. If
you want
the angle
use inverse
sine.