9/29

Test Wednesday

Pick up review & calculator

Have Motion WS II out

Warm up 2: Look at the V v T graph below. Draw the D v T graph
Make sure you
showing the same motion.
VvT
m/s
sec
write date,
question, &
answer for warm
ups. If you are
absent you must
get them from the
back.
10/1







Pick up calculator and trig notes.
Make sure calculator is in degree mode
Tue you handed in Motion WS II 1-12 & worked on Motion
Review
Wed you took the Motion Test
Test corrections/retakes: Thur pm, Mon am & pm, Tue am
ONLY
Warm Up #3
List 3 facts about triangles:
10/5
Today
you will finish Trig Packet
Tomorrow
No
I
you will have a quiz
warm up today
will be here after school 
11i)
0.633
10/6 YESTERDAY YOU FINISHED TRIG
PACKET. THERE WAS NOT A WARM
UP. QUIZ TODAY
Warm Up 4: Sketch the triangle
We will solve together
What do the 2 triangle have in common
Solve that first
12.68 cm
30cm
Now solve for X
12.98º
25º
X
55 cm
10/6 YESTERDAY YOU FINISHED TRIG
PACKET. THERE WAS NOT A WARM
UP. QUIZ TODAY
Warm Up 4: Solve for X
30cm
25º
X
55 cm
10/7 YESTERDAY YOU HAD A QUIZ. QUIZ CAN BE
MADE UP AFTER SCHOOL TODAY OR BEFORE
SCHOOL ON THURSDAY
 Pick
up Homework Set (Empire State) This is due
BOC Thursday
 Today
we will be going outside to do the survey
lab. (After we finish ex F of the application
notes)
 NOW:
 Trig
Test will be Friday Oct 16
 Warm


Have your notes out.
45º
Up #5
A
What is the missing angle?
What can we say
about sides A & B?
 TURN
X
IN WARM UPS TO SORTER
B
ANSWERS TO EMPIRE STATE
PROBLEM SET
1.
36.45º 28.07º
2.
335.14 ft
3.
18831.6 ft
4.
278.11 ft
 This
Monument is:
10/8 HAVE HW OUT

Yesterday we did a survey Lab. See me to make
it up. You also finished warm ups and turned
them in.

Today: Pick up Vector Note Sheet.

I have duty this pm until 2:50

Trig Quizzes should have been made up
yesterday or this morning.

HAPPY BIRTHDAY RANDI W!
What is the difference
between these tools?
We will be using a
triangulation device
like the bottom tool.
Note that the center
reference is 0º. This
allows us to get the
angle of elevation,
not the zenith angle.
SURVEY LAB
HOW WOULD I FIGURE OUT HEIGHT?
This is same as reading on
triangulation tool. If using a
45º

typical protractor, the
angle
would represent the zenith.
This describes the
angle of elevation
SURVEY LAB
HOW WOULD I FIGURE OUT HEIGHT?
This is same as elevation
angle since 90-45-45
triangle
45º

angle of elevation = 45º
SOH CAH TOA
opp
b
sin  

hyp
c
c  a b
2
2
2
adj
a
cos  

hyp c
opp b

tan  
adj a
TRIGONOMETRY REVIEW

We will be focusing
on triangles
What
is a right triangle?
A triangle with a 90º angle
What is a hypotenuse?
Side of right triangle opposite the 90º
angle
What is Pythagoreans Theorem?
c2 = a2 + b2
where c is the
hypotenuse.
Only applies to right triangles
EX A: GIVEN THE FOLLOWING TRIANGLE
a = 4.21u
b = 7.43 u
b
c
Angle C = 90.0°
What is the
hypotenuse (c) ?
a
EX A: GIVEN THE FOLLOWING TRIANGLE
c 2 = b 2 + a2
c2 = 4.212 + 7.432
b
c
a
c = 8.54 u
How would you
label the angles?
SAME TRIANGLE A
a = 4.21u
b = 7.43 u
c = 8.54 u
A
b
c
What is measure
of smallest angle,
θA?
θ is the Greek letter
a
theta and stands for
angle
This
is a good time to
review SOH CAH TOA
What does sine, cosine,
and tangent?
SOH CAH TOA
What
does sine, cosine, and
tangent represent?
The RATIO between given sides
of a right triangle in reference to
a specific angle.
SOH CAH TOA
Triangle Demo
THE RATIOS…..
Sine
= opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent
These only work for right triangles!
Show
Table
Angle SinA CosA TanA
NAMING THE SIDES
This side is opposite
our angle.
This is the longest side
— the hypotenuse.
H
O
A right angled
triangle
A
This side is adjacent
to our angle.
The angle we
are interested in.
NAMING THE SIDES
O
H = Hypotenuse
O = Opposite
A = Adjacent
NAMING THE SIDES
O
A
H
A
H
H
O
A
O
H = Hypotenuse
O = Opposite
A
H
A = Adjacent
H
A
O
O
EX B CONSIDER THIS TRIANGLE.
WHAT IS THE SINE RATIO?
O=
4cm
30°
Opposite/Hypotenuse gives us the Sine Ratio.
sin 30° = 4cm/8cm = 0.5.
If you enter Sin 30 in your calculator you should
get 0.5. Try it! (sin button is in the trig menu)
If the opposite side was 6 cm, what would the
hypotenuse be?
EX C CONSIDER THIS
TRIANGLE. WHAT IS THE
ANGLE?
O=
5cm
θ
Name the sides in reference to the angle
Determine which trig function to use
Sin = O/H
To determine angle you use the inverse trig function for
and enter the ratio of the corresponding sides.
Sin-1 (5/12) = 24.62º
Now go back to Example
A and solve the angle
using the inverse cosine
function, then solve the
angle using the inverse
tan function
a = 4.21u
b = 7.43 u
c = 8.54 u
What is measure of
smallest angle, θA?
A
b
c
EXAMPLE A
a
Cos θA = adj/hyp
Cos-1 θA (7.43/8.54)
θA = 29.54°
a = 4.21u
b = 7.43 u
c = 8.54 u
What is measure of
smallest angle, A?
A
c
b
EXAMPLE A
a
Tan θ = opp/adj
Tan-1 θA (4.21/7.43)
θ A = 29.54°
The
sum of all angles in a
triangle equals
180º
180º
- 90º - 29.54º
60.46º
HOW WOULD YOU
DETERMINE THE LAST ANGLE
B?
FOR RIGHT
TRIANGLES
If
you know any two sides,
you can determine the
angle
If you know a side and an
angle other than 90, you
can determine a side
EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U.
THE SMALLEST ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE
MEASURE OF SIDE S? WHAT IS THE MEASURE OF SIDE T? WHAT IS
THE MEASURE OF THE REMAINING ANGLE?
T
22º
S
28 u
EX D: A RIGHT TRIANGLE HAS A HYPOTENUSE MEASURING 28.0 U. THE SMALLEST
ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE
MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE?
Label Sides What do you know? Hyp and angle
What function can you use to solve for opp? Sin = Opp/Hyp
Opp = Sin Hyp
Opp = (Sin22º)(28u)
T adj
Opp = 10.49u
22º
S
28 u
opp
ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE
MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE?
c2 = a2 + b2
How would you solve for side T?
I will call adjacent (T) side a and opposite (S) side b a2 = c2 - b2
a2 = (28u)2 – (10.49u)2
T adj
a = 25.96u
22º
S
28 u
10.49 u
ANGLE HAS A MEASURE OF 22.0°. WHAT IS THE MEASURE OF SIDE S? WHAT IS THE
MEASURE OF SIDE T? WHAT IS THE MEASURE OF THE REMAINING ANGLE?
How would you solve for the remaining angle *?
Remember angles equal 180º
Angle * = 68º
180º – 90º - 22º
T adj
22º
S
28 u
*
10.49 u
Summary Putting it all together:
If you need to determine an angle :
 Name sides in
reference to angle of interest
 Determine formula
 You

know opp and hypotenuse, want θ :
sin-1 = (Opp/Hyp) Use inverse function
 sin-1 (5m/10m)=30º
10m hyp
5m opp
Θ ??
Summary Putting it all together:
If you need to determine a side:
 Name
sides in reference to known angle
 Determine
formula
You know angle and hypotenuse, want opposite:
Opp = (sinθ)(Hyp)
(sin30º)(10m) = 5m
10m hyp
?? opp
30º
Real World
Applications

EX E THE SWIMMER
A swimmer attempts to swim due north to the pier 2.00
miles away but the current takes him at a bearing of 40°.
After a while he notices he is due east of the pier. How far
has he travelled?
Step 1. Draw a diagram.
2.00 miles
pier
?
40°
EX E THE SWIMMER
Step 2. Identify the sides.
2 40° ?
Here we have the Adjacent
side and want to find the
Hypotenuse. So we use the
CAH triangle.
Putting our finger on H shows that H = A/C
= 2.00 ÷ (cos 40°)
=
= 2.61 miles
A
C H
EX F FINDING AN ANGLE (1)
At Heathwick airport there is a forest just 500. m from the
end of the runway. The trees can be as tall as 30. m. What
is the minimum angle of climb if aircraft are to avoid the
trees?
30.m
Step 1. Draw a diagram.
?
500.m
EX F FINDING AN ANGLE (2)
30
Step 2. Identify the sides

500
Here we have the Adjacent
and Opposite sides and want
to find an angle. So, we use
the TOA triangle.
Putting our finger on T shows
that… tan  = O/A
We can use the inverse tan to
find the angle.
 = tan-1 (30m/500m)
 = 3.4°
O
T A
EX G THE CHURCH STEEPLE
Eric decides to find the height of the steeple of his local
church. He measures a distance of 50. m along the ground.
The angle of elevation to the top of the steeple is 35°.
How high is the steeple?
Step 1. Draw a diagram.
?
50.m
35°
THE CHURCH STEEPLE
Step 2. Identify the sides.
?
35°
50
Here we have the Adjacent
side and want to find the
Opposite. So, we use the TOA
triangle.
Putting our finger on O shows
that O = T × A
= (tan (35º) × 50m
= 35.01 m
O
T A
REMEMBER…
A
C H
O
S H
O
T A
SOH-CAH-TOA
SIN
FINDING THE OPPOSITE
?
?
SOH-CAH-TOA
O
S H
O=
? cm
30°
Opp
= Sin  × Hyp
= (Sin 30°) × 8
= 4 cm
COS
FINDING THE ADJACENT
?
?
SOH-CAH-TOA
A
C H
27°
A = ? km
Adj
= Cos  × Hyp
= (Cos 27°) × 12.3
= 0.891 × 12.3
= 11.0 km
TAN
FINDING THE OPPOSITE
53°
?
?
SOH-CAH-TOA
O
T A
A=
16 cm
O = ? cm
Opp = Tan  × Adj
= (Tan 53°) × 16
= 1.327 × 16
= 21 cm
SIN
FINDING THE HYPOTENUSE
?
?
SOH-CAH-TOA
O
S H
O=
87 m
36°
Hyp
= Opp  Sin 
= 87  (Sin 36°)
= 87  0.5878
= 150 m
COS
FINDING THE HYPOTENUSE
60°
?
?
SOH-CAH-TOA
A
C H
A=
0.80 cm
Hyp
= Adj Cos 
= 0.80  (Cos 60.°)
= 0.80  0.50
= 1.6 cm
TAN
FINDING THE ADJACENT
?
?
SOH-CAH-TOA
O
T A
O=
3.1 cm
30°
A = ? cm
Adj
= Opp  Tan 
= 3.1  (Tan 30.°)
= 3.1  0.5773
= 5.4 cm
WHAT HAPPENS WHEN YOU DON’T KNOW THE
ANGLE?
We can find the usable number mentioned
previously using the ratios.
The problem is we know need to convert it
back into the original angle.
The Buttons on your calculator are…
Sin
Cos
Tan
The opposite of these are SHIFT then
Sin-1
Cos-1
Tan-1
SIN
FINDING THE ANGLE

?
?
?
SOH-CAH-TOA
O
S H
O = 3.0 km
Sin  = Opp  Hyp
Sin  = 3.0  7.0
Sin  = 0.4285

= Sin-1 (0.4285)

= 25°
COS
FINDING THE ANGLE
?
?
?
SOH-CAH-TOA
A
C H

A = 12.1 cm Cos  = Adj  Hyp
Cos  = 12.1  14.5
Cos  = 0.834

= Cos-1 (0.834)

= 33.4 °
TAN
FINDING THE ANGLE
?
?
?
SOH-CAH-TOA
O=
67.0 cm
O
T A

A = 187 cm Tan  = Opp  Adj
Tan  = 67.0  187
Tan  = 0.358

= Tan-1 (0.358)

= 19.7°
If
two vectors are not at right
angles to each other then we
must use the Law of Cosines:
C2 = A2 + B2 – 2AB cos 
“” or Theta, is any unknown
angle but in this case it is the
angle between the two
vectors