Romac
College
Algebra 2015
TRIGONOMETRY
Unit 9
Chapter 9*
Right Triangles
Geometric Mean
SOH – CAH – TOA
Summary Sheet
A2 + B2 = C2 
Adj2 + Opp2 = Hyp2
SIN θ = Opp/ Hyp
SOH  θ = ARCSIN (Opp/ Hyp)
COS θ = Adj/ Hyp
CAH  θ = ARCCOS (Adj/ Hyp)
TAN θ = Opp/ Adj
TOA  θ = ARCTAN (Opp/Adj)
A vector is a Right Triangle
X = Hyp * (COS(Ɵ))
Y = Hyp * (SIN(Ɵ))
c2 = a2 + b2 – 2ab*COS(C)
b2 = a2 + c2 – 2ac*COS(B)
a2 = c2 + b2 – 2cb*COS(A)
V * (COS(Ɵ))
SIN (A) / a = SIN (B) / b = SIN (C) / c
V* (SIN(Ɵ))
Vr = √ {Vx Tot2 + Vy Tot2}
Θrelative = ARCTAN (y Tot2 / Vx Tot)
Geometric Mean Are Ratios
BD/CD = CD/AD base to base
AB/BC = BC/BD hyp to base
AB/AC = AC/AD hyp to base
C
A
D
B
Unit 9 Lesson 1
At the end of this Unit 9 you will
be able to find
 All
three lengths of sides of a triangle
 All three angles inside a triangle
 Altitudes of triangles
 Areas of triangles
 How polygons can be formed by triangles


Lengths
Angles
Unit 9 Lesson 1
OBJECTIVE It is all ratios!:
- Use Similar Triangles to calculate
missing sides and angles of triangles. Pg
528 Ex1
-Calculate the Geometric Mean of a right
triangle pg 529 Ex2
C
A
D
B
Geometric Mean Are Ratios
BD/CD = CD/AD base to base
AB/BC = BC/BD hyp to base
AB/AC = AC/AD hyp to base
DRAW THIS OUT!!!!
DO NOW:
1) Kuda Handout
#’s 3 &7,
2) X/4 = 12/16 X =______
IN CLASS:
Page 528, 530
Examples 1, 2
Page 531 #’s 1 -7 ALL
Geometry Text Homework:
Page 531- 534
#’s 11-30 All
Honors 11- 29 ODD
+ 31, 32, 33
Pythagorean (triple)
A2 + B2 = C2
Radical = √
Area = ½ b*h
6
Unit 9 Lesson 2
IN CLASS:
Kuta Handout:
1 – 15 ODD
X
8
62 + 82 = x2
x = √ {36 + 64} = √100 = 10
6
10
X
62 + x2 = 102
x = √ {100 – 36} = √64 = 8
HOMEWORK:
Kuda Handout :
1 -15 ALL
Unit 9 Lesson 3
OBJECTIVES:
-Are they right triangles?
-Are they Acute or Obtuse?
-Are The Rectangles or Squares?
C
A
D
DO NOW:
For a RIGHT TRIANGLE
find the missing Side
62 + 82 = C2
C =___
10
12.6491
A2 + 62 = 142
A =___
Geometric Mean Are Ratios The Hypotenuse = 8
and one side = 4
BD/CD = CD/AD
√48 = 6.9282
Missing side = _____
AB/BC = BC/BD
AB/AC = AC/AD
IN CLASS:
LEFT Find the missing sides
Page 543-44 Examples 1, 2,3
B
Page 545: 1 – 7 ALL
If AB = 14, BC = 5 and AD=3
Find BD = ___
AB – AD = 11
Pythagorean = √33 = 5.7446
Find CD =____
Pythagorean = √42 = 6.4807
Find AC =_____
Geo Text copy Homework:
Page 546#’s 9 – 25 ODD
Unit 9 Lesson 4
- OBJECTIVE: Review
Geometric Mean
Pythagorean
Acute Or Obtuse
Geometric Mean Are Ratios
BD/CD = CD/AD
AB/BC = BC/BD
AB/AC = AC/AD
A2 + B2 = C2
Acute Largest Angle < 90
Or Obtuse Largest Angle > 90
DO NOW:
Do the three sides make a
right triangle?
20, 48, 56 Prove it
IN CLASS: USE GEO TEXT
Page 545: Example 3
#’s 1 – 7 all
Page 546
#’s 9 – 31 ODD
Homework:
Page 549 #’s 47 – 57 ALL
#’s 1- 8 Quiz 1
Unit 6 Lesson 5 9.4
- OBJECTIVE:
Special Right Triangles
45 – 45 – 90
60 – 30 – 90
Triangle Area
Page 551
45 – 45 – 90
Hyp = √(2) * Leg
Leg
DO NOW: Is this a right
triangle? Prove it.
2
Leg
Hyp
Page 551
Short
30 – 60 – 90
Leg
Hyp = (2) * Short Leg
OR
Long Leg = √(3) Short Leg
Long Leg
Hyp
5
3√3
IN CLASS:
Page 551 - 553
Examples 1 – 5
Page 554 #’s 1 – 11 all
Homework:
Practice “A” 1 – 15. ODD
Practice “B” 1 – 15 ODD
Quick Test
Use only your
Summary sheet
PRINTED
Given:
DE = 11.5
EY = 7.1589
HD = 13.0
E
Unit 6.5 Lesson 5 Jan 18 9.4
Find Lengths:
4.0000
HY = _________
8.2006
HE = _________
YD = _________
13.0000
Angle Measures
<EHY = _________
60.8059
<EDY = _________
38.4999
51.5001
<DEY = _________
29.1941
<HEY = _________
H
Y
D
Unit 9 Lesson 5
- OBJECTIVE:
Special Right Triangles
45 – 45 – 90
60 – 30 – 90
DO NOW: Is this a right
triangle? Prove it.
9
15
Triangle Area = ½ Base * Altitude
Page 551
45 – 45 – 90
Hyp = √(2) * Leg
Leg
Leg
Hyp
Page 551
Short
30 – 60 – 90
Leg
Hyp = (2) * Short Leg
OR
Long Leg = √(3) Short Leg
Long Leg
Hyp
12
IN CLASS:
Practice “A” 1 – 15 ALL
Homework:
Practice “A” 1 – 15. ALL
Practice “B” 1 – 15 ALL
Page 549 # 1- 8 Q
Unit 9 Lesson 8
Basic Trigonometric
Functions
Sine – Cosine – Tangent
DO NOW:
Handout # 1 & 2
IN CLASS:
Handout Page
#’s
Homework:
KUDA Worksheet
#’s 11 – 37 ODD
Unit Lesson 9
– SOH CAH TOA
FINDING THE SIDES
DO NOW:
Hyp
 Given
a Right Triangle with
one angle and one side


Find the missing angles and
missing side
A2 + B2 = C2 




Adj2 + Opp2 = Hyp2
SIN θ = Opp/ Hyp
COS θ = Adj/ Hyp
TAN θ = Opp/ Adj
SOH
CAH
TOA
Opp
Θ
Adj
IN CLASS:
Page
#’s
Homework:
KUDA Worksheet REDO
#’s 10 – 38 EVEN
Unit 9 Lesson 10
The Pythagorean Theorem is a
corollary of the Law of COSINES**.
AC = 21
BC = 12
<c = 40 deg
Unit 9 Lesson 10-a
Law of COSINE
 c2
= a2 + b2 – 2ab*COS(C)
 b2 = a2 + c2 – 2ac*COS(B)
 a2 = c2 + b2 – 2cb*COS(A)
B
a
C
A
b
Law of SINE
SIN (A) / a = SIN (B) / b = SIN (C) / c
C
A
D
Review Geometric Mean Ratios
BD/CD = CD/AD
AB/BC = BC/BD
AB/AC = AC/AD
B
c
Unit 9 Lesson 8 Trig uses
– #1 VECTORS
DO NOW:
Hyp
A
vector is a right Triangle
 X = Hyp * (COS(Ɵ))
 Y = Hyp * (SIN(Ɵ))
 phet-vector-addition
 calculator
IN CLASS:
Page
#’s
Opp
Θ
Homework:
KUDA work sheet
TBA
Adj
Unit 9 Lesson 8a
– VECTORS

A vector is a right Triangle

X = Hyp * (COS(Ɵ))
Y = Hyp * (SIN(Ɵ))



Vr = √{Xtotal2 + Ytotal2 }
Ɵ = ARCTAN ( Ytotal / Xtotal )
Vr* (COS(Ɵ))
Vr* (SIN(Ɵ))
100* (COS(45))
70.7107
100* (SIN(45))
71.7107
50* (COS(120))
-25.0000
50* (SIN(120))
43.3013
25* (COS(270))
0.0000
25* (SIN(270))
-25.0000
75* (COS(195))
-72.4444
75* (SIN(195))
-19.4114
Xtotal = – 26.7337
Ytotal = + 70.6006
DO NOW:
What is the Horizontal (X) component
of a Vector measuring 10 meters /
second at 25 degrees from the
horizon?
X = Hyp * (COS(Ɵ))
X = 10 * (COS(25))
= 9.0631
Last Nights Homework:
V1 = 100 @ 45 degrees
V2 = 50 @ 120 degrees
V3 = 25@ 270 degrees
V4 = 75@ 195 degrees
Vr = √{Xtotal2 + Ytotal2 } = √{5699.1354}
Vr = 75.4926
Ɵ = ARCTAN ( Ytotal / Xtotal )
Ɵ = 69.2603 degrees RELATIVE
Ɵ = – 20.7421 Actual
Unit 9 Lesson 9a
– VECTORS

X comp = Vr* (COS(Ɵ))  Horizontal
Y comp = Vr* (SIN(Ɵ))  Vertical

Vr = √ {Vx Tot2 + Vy Tot2}




Θ = ARCTAN ( Vy / Vx)
DO NOW
V1 = 5 @ 45º
V2 = 4 @ -30º
Vr* (COS(Ɵ))
Do NOW: Find the resultant
of V1: 5@45 degrees
and V2 : 4 @ -30 degrees
IN CLASS: Do “By Table
Team”- On board
Vr* (SIN(Ɵ))
Vx1 = 5 COS (45º)
= 3.5355
Vy1 = 5 SIN (45º)
= 3.5355
Vx2 = 4 COS (330º)
= 3.4641
Vy2 = 4 SIN (330º)
= -2.0000
Vx Total = 6.9996
Vy Total = 1.5355
Vr = √ {Vx Tot2 + Vy Tot2} = √ {6.99962 + 1.5355 2}
Vr = 7.1660
Θ = ARCTAN ( Vy / Vx) = ARCTAN (1.5355 / 6.9996)
Θ = 12.3730
IN CLASS:
Find the resultant of four Vectors
IF Vector 1 = 10 @ 20º
QI
IF Vector 2 = 20 @ 120º QII
IF Vector 3 = 30 @ 220º QIII
IF Vector 4 = 40 @ 320º QIV
Addition of Vectors – Graphical Methods
Even if the vectors are not at right angles, they can be added
graphically by using the “Head
to Tail” method:
1. Draw V1 & V2 to scale.
2. Place tail of V2 at tip of V1
3. Draw arrow from tail of V1 to tip of V2
This arrow is the resultant V
(measure length and the angle it makes with the x-axis)
Same for any number of vectors involved.
Unit Lesson 9a
Practice
Vr* (COS(Ɵ))
Vr* (SIN(Ɵ))
Vx1 = 10 COS (20º) Vy1 = 10 SIN (20º)
= 9.3969
= 3.42092
Vx2 = 20 COS (120º) Vy2 = 20 SIN (120º)
= -10.0000
= 17. 3205
Vx3 = 30 COS (220º) Vy3 = 30 SIN (220º)
= – 22.9813
= – 19.2836
Vx4 = 40 COS (320º) Vy4 = 40 SIN (320º)
= 30.6419
= – 25.7115
Vx Total = 7.0574 Vy Total = – 24.2544
IN-CLASS:
Find the resultant of FOUR Vectors
IF Vector 1 = 10 @ 20º
IF Vector 2 = 20 @ 120º
IF Vector 3 = 30 @ 220º
IF Vector 4 = 40 @ 320º
Vr = √ {Vx Tot2 + Vy Tot2} =
√ [7.0574 2 + {– 24.25442} ]
Vr = √ [638.0828] = 25.2603
Θ = ARCTAN ( Vy / Vx) =
ARCTAN (24.2544 / 7.0574)
Θrel = 73.8417
Θreal = 360 – 73.8417
= 286.1503
ANSWER = 25.2603 @ 286.1503 degrees
Unit 9 Lesson 9a
Practice
Vr* (COS(Ɵ))
Vr* (SIN(Ɵ))
Vx1 = 30 COS (10º) Vy1 = 30 SIN (10º)
= 29.5442
= 5.2094
Vx2 = 12 COS (150º) Vy2 = 12 SIN (150º)
= – 10.3923
= 6.0000
Homework :
Find the resultant of 4 Vectors
IF Vector 1 = 30 @ 10º
IF Vector 2 = 12 @ 150º
IF Vector 3 = 25 @ 350º
IF Vector 4 = 11 @ 190º
Vr = √ {Vx Tot2 + Vy Tot2} = √
[32.9392 2 + 4.95812} ]
Vr = √ [1109.5737] = 33.3103
Vx3 = 25 COS (350º) Vy3 = 25 SIN (350º)
= 24.6202
= – 4.3412
Vx4 = 11 COS (190º) Vy4 = 11 SIN (190º)
= – 10.8329
= 1 – 1.9101
Θ = ARCTAN ( Vy / Vx) =
ARCTAN (4.9581/ 32.9392)
Θrel = 8.5601 degrees
Θreal = 8.5601 degrees
Vx Total = 32.9392 Vy Total = 4.9581
ANSWER = 33.3103 @ 8.5601 degrees
Unit 9 Lesson 10
Review SIN – COS – TAN
Review Distance Formula
DO NOW:
Dist = √{ (x2 – x1)2 + (y2 – y1)2}
Vector Components:
‹ x, y, ›
Compass vs. Cartesian





COSINE FUNCTION
SINE FUNCTION
TANGENT FUNCTION
phet-vector-addition
calculator
IN CLASS:
Page 573 Example’s 2- 5
Page 576 #’s 1 – 9 all
Homework:
Page 576
#’s 10 – 28 EVEN
If angle Θ = 350 degrees
And V = 10
Find “Vertical” and
“Horizontal” components
Vertical = V * (SIN(Θ))
Horizontal = V * (COS(Θ))
Vertical = 10(SIN350))
Horizontal = 10(COS(350))
Vertical = – 1.7365
Horizontal = + 9.8481
Unit 9 Lesson 10
Lesson 10 Homework/quiz
Compass  Cartesian Compass



Hopes’ mom, is a Southwest Airlines Captain. She
flies several versions of the Boeing 737 jetliner.
Thursday she flew for Orlando Florida on a heading
of 030 degrees Compass at 420 knots (nautical
miles per hour). The wind was blowing at 50 knots
in a southeastern direction (110 degrees compass).
Without correction what would her track over the
ground be?
Speed and direction (in compass)
Extra credit: What course would she have to fly to
actually reach Boston with a sustained 50 knot
wind?
Cartesian vs. Compass
NORTH – 360 /000
EAST - 090
WEST- 270
SOUTH - 180
Unit 9 Lesson 12
Review
The New-New Dorm has a roof Truss
pictured in the figure to the right.
If AB = 42, BC = 24 and AC=20
Find BD = ___
Find CD =____
A
Find AD =_____
Find the AREA of the Triangle formed by the truss = ________
C

D
Romac is rowing his boat across the Kennebec River. He can row
at 6.0 miles per hour due east (090 compass - 0.00 degrees). The
river current is 2.5 miles per hour heading due south (180 compass 270.0 degrees). Finally the wind is blowing the boat at 1.5 miles per
hour at northeast
(045 compass – 45 degrees.)
What is the resultant velocity = _____________
and direction compass = ______________
of the boat when the three vectors work on it?


Honors if the river is 1.0 miles across how long will it take Romac to
reach the other side?

Find the Equation of the line that passes through the coordinates (-2, 4)
and (-3, 5) in the
B
Summary Sheet
A2 + B2 = C2 
Adj2 + Opp2 = Hyp2
SIN θ = Opp/ Hyp
SOH  θ = ARCSIN (Opp/ Hyp)
COS θ = Adj/ Hyp
CAH  θ = ARCCOS (Adj/ Hyp)
TAN θ = Opp/ Adj
TOA  θ = ARCTAN (Opp/Adj)
A vector is a Right Triangle
X = Hyp * (COS(Ɵ))
Y = Hyp * (SIN(Ɵ))
c2 = a2 + b2 – 2ab*COS(C)
b2 = a2 + c2 – 2ac*COS(B)
a2 = c2 + b2 – 2cb*COS(A)
V * (COS(Ɵ))
SIN (A) / a = SIN (B) / b = SIN (C) / c
V* (SIN(Ɵ))
Vr = √ {Vx Tot2 + Vy Tot2}
Θrelative = ARCTAN (y Tot2 / Vx Tot)
Geometric Mean Are Ratios
BD/CD = CD/AD base to base
AB/BC = BC/BD hyp to base
AB/AC = AC/AD hyp to base
C
A
D
B
Quick Test
Use only your
Summary sheet
PRINTED
Given:
DE = 11.5
EY = 7.1589
HD = 13.0
E
Feb 19
Find Lengths:
4.0000
HY = _________
8.2006
HE = _________
YD = _________
9.0000
Angle Measures
<EHY = _________
60.8059
<EDY = _________
38.4999
51.5001
<DEY = _________
29.1941
<HEY = _________
H
Y
D
Unit 9 Lesson 9a FEB 5th
Practice
Vr* (COS(Ɵ))
Vr* (SIN(Ɵ))
Vx1 = 10 COS (20º) Vy1 = 10 SIN (20º)
= 9.3969
= 3.42092
Vx2 = 20 COS (120º) Vy2 = 20 SIN (120º)
= -10.0000
= 17. 3205
Vx3 = 30 COS (220º) Vy3 = 30 SIN (220º)
= – 22.9813
= – 19.2836
Vx4 = 40 COS (320º) Vy4 = 40 SIN (320º)
= 30.6419
= – 25.7115
Vx Total = 7.0574 Vy Total = – 24.2544
IN-CLASS:
Find the resultant of FOUR Vectors
IF Vector 1 = 10 @ 20º
IF Vector 2 = 20 @ 120º
IF Vector 3 = 30 @ 220º
IF Vector 4 = 40 @ 320º
Vr = √ {Vx Tot2 + Vy Tot2} =
√ [7.0574 2 + {– 24.25442} ]
Vr = √ [638.0828] = 25.2603
Θ = ARCTAN ( Vy / Vx) =
ARCTAN (24.2544 / 7.0574)
Θrel = 73.8417
Θreal = 360 – 73.8417
= 286.1503
ANSWER = 25.2603 @ 286.1503 degrees