Transcript c - udcompsci


c
a

b
To solve a right triangle means to
find the missing lengths of its sides
and the measurements of its angles.
Given Right Triangle ABC,
where a = 4 and A = 35, solve the triangle.
B

4
4
tan 35 
b
b  4 / tan 35  5.71
c

35
b
4

sin 35 
c
c  4 / sin 35  6.97
C
A
  90  35  55



Given Right Triangle ABC,
where c = 9 and b = 2, solve the triangle.
B

9
a

C
2
A
A 25 foot ladder is leaning against a wall and forms
an angle of 70 degrees with the ground. How high
up the wall is the top of the ladder?
h
sin70 
25

h
25

70

h  25 sin 70
h  235
. feet
Indirect Measurement using
Angles of Elevation and Depression
Angle of Elevation: an angle whose initial side is some
horizontal (usually the ground) and whose terminal
side elevates (or rises) from that horizontal.
Angle of Depression: an angle whose initial side is some
horizontal and the terminal side falls (or depresses) from
that horizontal.
1. From a point on the ground 120 meters away from the foot of
the Eiffel Tower, the angle of elevation of the top of the
tower is 68.2. How high is the tower?
Solving Oblique Triangles
Oblique Triangles are ones that do NOT have
a right angle.
To solve an oblique triangle means to
determine the measures of all the missing
angles and the lengths of all the missing
sides.
We use either the Law of Sines or the Law of
Cosines to solve Oblique Triangles.
Whether we use the Law of Sines or the Law of Cosines
depends on what information we are given about the triangle.
CASE 1: One side and two angles are known (SAA or
ASA).
CASE 2: Two sides and the angle opposite one of them
are known (SSA-The Ambiguous Case).
CASE 3: Two sides and the included angle are known
(SAS).
CASE 4: Three sides are known (SSS).
We use the Law of Sines for Cases 1 and 2, and the Law of Cosines for Cases 3 and 4
A
S
A
ASA
S
A
A
SAA
CASE 1: ASA or SAA
S
A
S
CASE 2: SSA
S
A
S
CASE 3: SAS
S
S
S
CASE 4: SSS
Law of Sines
For a triangle with sides a , b, c and opposite
angles  ,  , , respectively,
sin  sin  sin 


a
b
c
Lesson Overview 5-6B
Example: Given Triangle ABC with side a = 12,
angle A = 36 degrees, and angle B = 22 degrees,
solve the triangle.
Example: Given Triangle ABC with side b = 56,
angle A = 29 degrees, and angle C = 104 degrees,
solve the triangle.
Lesson Overview 5-7A
Lesson Overview 5-7B
Example: Given Triangle ABC with side b = 3,
side c = 5, and angle B = 50 degrees,
solve the triangle.
Example: Given Triangle ABC with side b = 5,
angle side c = 3, and angle B = 30 degrees,
solve the triangle.
Example: Given Triangle ABC with side b = 8,
angle side c = 10, and angle B = 45 degrees,
solve the triangle.
Theorem Law of Cosines
For a triangle with sides a , b, c and opposite
angles  ,  , , respectively.
c  a  b  2ab cos
2
2
2
b  a  c  2ac cos 
2
2
2
a  b  c  2bc cos
2
2
2
Example: Given Triangle ABC with side b = 6,
angle side a = 9, and angle C = 37 degrees,
solve the triangle.
Example: Given Triangle ABC with side b = 8,
angle side c = 7, and side a = 9 degrees,
solve the triangle.