Transcript Trigonometry_partII

Trigonometry Part II
Math 416
Game Plan
 Area of Triangles
 Traditional
 Sine
 Hero’s
 Sine Law
 Two examples
 Word Problem
Traditional Area of Triangles
 We are now going to be working with other
triangles besides right angle triangles
 The 1st concept we will look at is area
 Area of triangle = half the base x height
 A = ½ b h or A = bh
2
Using Sin for Area of Triangle
 We note the base is a side of the triangle
 The height must be at 90° or perpendicular
to that base (not up and down)
Note b =8
 Eg
9
8
h =9
A = ½ bh
A = ½ (8)(9)
A = 36
Sin
 Now let us try and get a different
perspective.
 Consider
We are given two sides
and the contained angle.
From trig Sin θ = h
p
p
h = p Sin θ
θ
Create
formula for A
q
From formula A = ½ bh
A = ½ qp Sinθ
Another Example
Find Area
12
A = ½ (18)(12) Sin 47°
47°
18
A = 78.99
Using Hero’s to find Area of
Triangle
 Now a totally different approach was
found by Hero or Heron
 His approach is based on perimeter
of a triangle
Be My Hero and Find the Area
 Consider
P = a + b + c (perimeter)
p = a + b + c / 2 or
p = P / 2 (semi perimeter)
b A = p (p-a) (p-b) (p-c)
a
c
Hence, by knowing
the sides of a
triangle, you can find
the area
Be My Hero and Find the Area
 Eg
P = 9 + 11 + 8 = 28
p = 14
A = p (p-a) (p-b) (p-c)
9
11 A = 14(14-9)(14-11)(14-8)
A = 14 (5) (3) (6)
8
A = 1260
A = 35.5
Be My Hero and Find the Area
P = 42 + 43 + 47
p = 66
A = p (p-a) (p-b) (p-c)
 Eg
42
43 A = 66(24)(23)(19)
47
A = 692208
A = 831.99
Be My Hero and Find the Area
P=9+7+3
p = 9.5
A = p (p-a) (p-b) (p-c)
 Eg
9
7
3
A = 9.5(0.5)(2.5)(6.5)
A = 77.19
A = 8.79
Sin Law
With respect to Angle A
 Consider
Sin A = h/b
h = b sin A
C
With respect to Angle B
Sin B = h/a
h
=
a
sin
B
b
a
h
Thus, aSinB = bSinA
Divide both sides by ab
B
A Sin B = Sin A
b
a
Sin Law
 Now we can do this again using Angle C
 What we get is the Sin Law for side
lengths
 a
= b
= c
Sin A
Sin B
Sin C
Sin Law for angles
 Sin A = Sin B = Sin C
a
b
c Notice when getting
angles Sin on TOP
(think a on top)
.
.
Notes
 Each expression is actually 3 formulae
 You do not need the whole thing
 Always look for the Side – Angle - Combo
1st Example
 Eg
8
57°
β
Complete the Triangle
Let’s get angle or θ 1st
15
θ
Sin θ = Sin 57
8
15
x
θ = 27°… now Beta
β = 180 – 57 – 27 = 96°… now x
x = 15
x = 17.79
Sin 96 Sin 57
2nd Example
 Eg
Complete the Triangle
st
Let’s
get
θ
1
θ x
θ = 180 – 75 = 62… x?
y
x =
18
75°
43°
Sin 75 Sin 62
18
x = 19.69… now y?
y =
18
y
=
13.90
Sin 43 Sin 62
Word Problem
 A surveyor creates the following map
x =
200
Sin 73 Sin 63
Billy’s House
x = 214.66
y =
200
Sin 44 Sin 63
200m y = 155.93
Dist
=
63°
73°
214.66+155.93+200
School
Post Office = 570.59 m
 What is the shortest distance if Billy goes from
home to school, to the post office and home?