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Metric Relations in Right Triangles
• By drawing the altitude
from the right angle of a
right triangle, three similar
right triangles are formed
Geometric Properties
• Using the lengths of the corresponding sides
of the triangles formed, we can determine the
ratios and from this determine certain
geometric properties
Property 1
• In a right triangle the length
of the leg of a right triangle
is the geometric mean
between the length of its
projection on the
hypotenuse
A.)
• In easiest terms, a leg
squared is equal to the
hypotenuse multiplied by
the leg’s projection on the
hypotenuse.
Property 2:
• The square of the altitude is
equal to one part of the
hypotenuse multiplied by
the other
Property 3:
• In a right triangle, the
product of the length of the
hypotenuse and its
corresponding altitude is
equal to the product of the
lengths of the legs.
Class Work and Homework
• Visions P. 116 # 9
• Visions P. 115 # 2-8, 10, 13, 15, 19
• Math 3000 P. 217 # 1, 3, 4
Trigonometry
Definition:
• Trigonometry is the art of studying triangles
(in particular, but not limited to, right
triangles)
• Deals with the relationships between the
angles and side lengths of a triangle
• Before learning the key formulas in
trigonometry, it is of absolute importance that
some terms are understood
• Because we are dealing with right triangles,
you are already familiar with one very
important right triangle theorem:
– The Pythagorean Theorem
a² + b² = c²
• In every right triangle, because one of the angles
measures 90°, then logically the other two angles must
add up to 90°
A
B
C
Because m<B = 90° then m<A + m<C = 90° (since there are
180° in every triangle)
Hypotenuse:
– The side that is opposite the
right angle
– The longest side in the right
triangle
A
Opposite Side:
– The side that is opposite of a
given angle
– Ex: Side AB is opposite m<C
Side BC is opposite m<A
Adjacent Side:
– The side that is neither the
hypotenuse or opposite
Ex: Side BC is adjacent to m<C
Side AB is adjacent to m<A
B
C
Example:
Fill in the side that corresponds to the following
questions:
A
Hypotenuse: _________________
Opposite m<A: _________________
Adjacent m<A: _________________
Opposite m<C: __________________
Adjacent m<C: __________________
B
C
These three definitions of the sides are of
utmost importance in trigonometry
They are at the root of finding every angle in a
right triangle
The MOST important Gibberish word you
will need to remember in math life
SOH – CAH - TOA
Opposite
sin A 
Hypotenuse
A
Adjacent
cos A 
Hypotenuse
Opposite
tan A 
Adjacent
B
C
Example:
opposite
1
sin 30 

hypotenuse
2
adjacent
3
cos 30 

hypotenuse
2
opposite
1
tan 30 

adjacent
3
3
sin 60 
2
1
cos 60 
2
3
tan 60 
1
A
60°
2
1
30°
B
C
3
Using Your Calculator
1. The keys sin, cos, tan on the calculator
enable you to calculate the value of sin A,
cos A, or tan A knowing the measure of
angle A
So if you know the measure of an angle you can
use the sin, cos, or tan buttons on your
calculator in order to calculate its value
2. The keys sin-1, cos-1, tan-1 on the calculator
enable you to calculate the measure of angle
A knowing sin A
• So if know sin A, cos A, or tan A, you can
calculate the measure of angle A
Homework
• Hand outs 1 and 2: Trigonometric Ratios and
Calculator
• Math 3000 pages 228, 229 # 2,3,4,5
P. 182 # 1
P. 184 # 2
Finding Missing Sides Using
Trigonometric Ratios
Finding the Missing Side of a Right Triangle
• In order to find a missing side, you will need
two pieces of information:
– An angle
– One side length
In a right triangle
1. Finding the measure x of side BC opposite to
the known angle A, knowing also the
measure of the hypotenuse, requires the use
of sin A
Remember: SOH
*****Cross Multiply*****
Sin 50º=
x
5
x=5sin50º = 3.83 cm
Finding the measure y of side AC adjacent to the
known Angle A, knowing also the measure of
the hypotenuse, requires the use of cos A
Remember: cos = adjacent/hypotenuse
*****Cross Multiply*****
y = 5 cos 50º = 3.21 cm
Cos 50º =
y
5
3. Finding the measure x of side BC opposite to
the known angle A, knowing also the
measure of the adjacent side to angle A,
requires the use of tan A
remember tan=opposite/adjacent
***cross multiply***
x
tan 30º =
4
x = 4 tan 30º = 2.31 cm
Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²
Sin( A) opp

1
hyp
Cos( A) adj

1
hyp
Tan( A) opp

1
adj
 opp 
A  sin 1 

 hyp 
 adj 
A  cos 1 

 hyp 
 opp 
A  tan 1 

 adj 
Example
In the following triangle, find the value of side x
(Remember to use SOH-CAH-TOA)!!!
x
sin 50 
5
5 sin 50  x
3.83  x
50
5cm
x
Example
In the following triangle, find the value of side y
(Remember to use SOH-CAH-TOA)!!!
x
cos 50 
5
5 cos 50  x
3.21  x
50
5cm
x
Example
• Find the value of x
8
tan 60 
x
8
x
tan 60
x  4.6
8
x
60
Class work and homework
• Finding missing sides
• math 3000: page 229 # 6,7
• Hand out
Finding Missing Angles using
Trigonometry Ratios
In a Right Triangle
1. Find the acute angle A when its opposite side
and the hypotenuse are known requires the
use of sin A
SOH – Opposite/hypotenuse
sin A =
4
5
M<A=sin-1 ( )=53.1º
4
5
2. Finding the acute angle A when its adjacent
side and the hypotenuse are known requires
the use of cos A
Cos = adjacent/hypotenuse
Cos A=
3
4
m<A = cos-1 ( ) = 41.4º
3
4
3. Finding the acute angle A when its opposite
side and adjacent side are known requires
the use of tan A
tan = opposite/adjacent
Tan A =
3
2
m<A=tan-1 ( ) = 56.3º
3
2
Find the missing Angle
• If we take the inverse of each formula, we can
find the missing side angle in a 90°
triangle
• The symbol for the inverse of
sin (A) is sin-1; cos (A) is cos-1;
tan (A) is tan-1
Formulas for 90° triangle
Formulas to find a missing side
Formulas to find a missing angle
(hyp)² = (opp)² + (adj)²
Sin( A) opp

1
hyp
Cos( A) adj

1
hyp
Tan( A) opp

1
adj
 opp 
A  sin 1 

 hyp 
 adj 
A  cos 1 

 hyp 
 opp 
A  tan 1 

 adj 
Example
sin 30º = 0.5 and sin-1 (0.5) = 30º
Class work and homework
Hand out Find the missing angles using trig
ratios
Math 3000 page 231 # 8,9
Math 3000 pg 232 # 10-15
10. x=10.4 cm; y=3.0 cm; z=9.0 cm; t=5.2 cm
11. ab=10cos64 = 4.38 cm; bc=10sin64= 8.99 cm; area: 39.4
cm squared
12. H=50tan40 = 42 m
13. Length of shadow = 50tan30 = 28.9 m
14. X=65tan54 degrees = 89.5 m
9.
c)
d)
e)
f)
B). c=50 degrees, ab=4,6, ac=3.9
mc=60 decrees, ab=5.2, ac=3
Bc=10, angle b=52.1 degrees, angle c=36.9 degrees
Ac=12, angle b=67.4, angle c=22.6
Ab=15, angle b=28.1 degrees, angle c=61.9 degrees
Solving a triangle
To determine the measure of all its
sides and angles
Sine Law
• The sides in a triangle are directly
proportional to the sine of the opposite angles
to these sides
a
b
c


sin A sin B sin C
• The sine law can be
used to find the
measure of a missing
side or angle
1st Case
• Finding a side when we know two angles
and a side
We calculate the measure x of AC
x
15
15sin 50


x 
 13.27cm
sin 50 sin 60
sin 60
How to:
1. Place Measurement x over sin known angle
2. Equal to
3. Measurement known side over sin of known
angle
4. Cross multiply and divide to find unknown
measurement
5. Calculate.
2nd Case
• Finding an angle when we know two sides
and the opposite angle to one of these two
sides
• We calculate the measure of angle B
10
13
10sin 50

sin B 
 0.5893 mB  36
sin B sin 50
13
• Make sure you have opposite angles and side
measurements. Remember total inside angles
must equal 180º
How to calculate if need to find an
angle:
1. Place side measurement known over sin of
angle we wish to know
2. Equal to
3. side measurement over sin angle we know
4. Cross multiply and divide to find x
5. To calculate angle –sin x = angle. Don’t forget
unit i.e.º
The sine of an obtuse angle
• The trigonometric
functions (sine, cosine,
etc.) are defined in a
right triangle in terms of
an acute angle. What,
then, shall we mean by
the sine of an obtuse
angle ABC?
• The sine of an obtuse angle is
defined to be the sine of its
supplement.
• How to find the measure of the degree of an
obtuse angle:
• Follow the procedure you have learned so far,
then subtract that angle from 180º
10 cm
22º
18.6 cm
10

sin 22 sin w
18.6
18.6sin 22
10
 .697
  sin .697
 44.2
 180  44.2  135.5
in
Class assignment
• Find all missing side lengths and angles and
math 3000 page 232 # 10-15
Area of a Triangle
There are three formulas to find the
area of any triangle
• The first of the three you are already aware of:
A=bxh
2
Area of a Triangle knowing Two Sides
and the Angle in Between
Area = ac ● sinB
2
A
c
Area = ab ● sinC
2
Area = bc ● sinA
2
B
b
a
C
• These are the 3 formulas that involve the sine of
an angle and the two sides that contain the
angle
Example:
- Find the area of the following triangle
6cm
Area = ac●sinB
2
Area = 6 ●12sin(36.3°)
2
Area = 21.3cm²
36.3°
12cm
Hero’s Formula
Area 
p( p  a)( p  b)( p  c)
Where ‘p’ = HALF of the perimeter
Example:
p = 6 + 8 + 12 = 26= 13
2
2
8cm
6cm
Area  13(13  6)(13  8)(13  12)
Area  13(7)(5)(1)
Area  455  21.3cm 2
12cm
Step 1: Label the sides of the triangle if relevant
Step 2: State what information we know
Step 3: Select a theorem and find the area
• In easiest terms, you need two side lengths
and the angle in between them to find the
area of any triangle
Example:
- Find the area of the following triangle
6cm
Area = ac●sinB
2
Area = 6 ●12sin(36.3°)
2
Area = 21.3cm²
36.3°
12cm
Area of a Triangle Knowing Two
Angles and One Side
Step 1: Draw the Altitude AD
Step 2: Find mAD
Step 3: Find mBD
10cm
Step 4: Find mDC
B
A
55
25
D
Step 5: Add mBD and mDC
Step 6: Use any formula you wish to find the
total area
C
Class work and homework
• Math 3000 pg 233 act 1 and 2, and pg 235
numbers 1-5, 7 and hand outs