6.16.1 the law of sines. quick review quick review solutions
TRANSCRIPT
Quick Review
Given / / , solve for the given variable.
1.
2.
Evaluate the expression.
8sin 323.
5Solve for the angle .
4. sin 0.2 0 90
5. sin 0.2 90 180
a b c d
b
c
x
x x
x x
Quick Review Solutions
Given / / , solve for the given variable.
1.
2.
Evaluate the expression.
8sin 323.
5Solve for the angle .
4. sin 0.2 0 90
0.848
11.53
5. sin 0.
7
162 90 180
a b c d
b
c
x
x x
ad
cad
x x
b
8.463
What you’ll learn about• Deriving the Law of Sines• Solving Triangles (AAS, ASA)• The Ambiguous Case (SSA)• Applications
… and whyThe Law of Sines is a powerful extension of
the triangle congruence theorems of Euclidean geometry.
Law of Sines
In with angles , , and opposite sides , , and , respectively,
the following equation is true:
sin sin sin.
ABC A B C a b c
A B C
a b c
Example Solving a Triangle Given Two Angles and a Side
Solve given that 38 , 46 , and 9.ABC A B a
Find 180 38 46 96 .
Apply the Law of Sines:
sin sin sin sin
sin38 sin 46 sin38 sin96
9 99sin 46 9sin96
sin38 sin38
10.516
C
A B A C
a b a c
b c
b c
b
14.538
The six parts of the triangle are:
38 9
46 10.516
96 14.538
c
A a
B b
C c
Example Solving a Triangle Given Two Sides and an Angle (The Ambiguous
Case)
Solve given that 7, 6, and 30 .ABC a b A
Example Solving a Triangle Given Two Sides and an
Angle (The Ambiguous Case)
Solve given that 7, 6, and 30 .ABC a b A
1
Use the Law of Sines to find .
sin30 sin
7 66sin30
sin7
25.4 or 180 25.4 154.6
If 25.4 , If
B
B
B
B B
B
154.6 ,
then 180 30 25.4 124.6 then 180 30 154.6 4.6
7sin124.6and 11.5 Since this is not possible,
sin30
B
C C
c
there is only one triangle.
The six parts of the triangle are:
30 , 7, 25.4 , 6, 124.6 , 11.5.A a B b C c
Example Finding the Height of a Pole
A road slopes 15 above the horizontal, and a vertical telephone pole
stands beside the road. The angle of elevation of the Sun is 65 , and
the pole casts a 15 foot shadow downhill along the road. Fin
d the height
of the pole.
x
15ft15º
65º
B
A
C
Example Finding the Height of a Pole
A road slopes 15 above the horizontal, and a vertical telephone pole
stands beside the road. The angle of elevation of the Sun is 65 , and
the pole casts a 15 foot shadow downhill along the road. Fin
d the height
of the pole.
x
15ft15º
65º
B
A
C
Let the height of the pole.
180 90 65 25
65 15 50
sin 25 sin50
1515sin50
27.2sin 25
The height of the pole is about 27.2 feet.
x
BAC
ACB
x
x
Quick Review
2 2 2
2 2
Find an angle between 0 and 180 that is a solution to the equation.
1. cos 4 / 5
2. cos -0.25
Solve the equation (in terms of and ) for (a) cos and
(b) , 0 180 .
3. 7 2 cos
4. 4
A
A
x y A
A A
x y xy A
y x
4 cos
5. Find a quadratic polynomial with real coefficients that has no real zeros.
x A
Quick Review Solutions
2
Find an angle between 0 and 180 that is a solution to the equation.
1. cos 4 / 5
2. cos -0.25
Solve the equation (in terms of and ) for (a) cos and
(b) , 0 180 .
3.
36.87
104.48
7
A
A
x y A
A A
2 2 2 2
-12 2
2
2 2 2 2
-12
49 49(a) (b) cos
2 2
4 4(a) (b) cos
2 cos
4. 4 4 cos
5. Find a quadratic polynomial with real coefficients that has no
4
ea
4
r
x y xy A
y x x
x y x y
xy xy
y x y x
x xA
2One answer
l ze
:
r s.
2
o
x
What you’ll learn about• Deriving the Law of Cosines• Solving Triangles (SAS, SSS)• Triangle Area and Heron’s Formula• Applications
… and whyThe Law of Cosines is an important
extension of the Pythagorean theorem, with many applications.
Law of Cosines
2 2 2
2 2 2
2 2 2
Let be any triangle with sides and angles
labeled in the usual way. Then
2 cos
2 cos
2 cos
ABC
a b c bc A
b a c ac B
c a b ab C
Example Solving a Triangle (SAS)
Solve given that 10, 4 and 25 .ABC a b C
2 2 2
2
2
Use the Law of Cosines:
2 cos
16 100 2(4)(10)cos25
6.6
Use the Law of Cosines again:
10 16 43.56 2(4)(6.6)cos
cos 0.7659
140
180 140 25 15
The six parts of the triangle are:
14
c a b ab C
c
c
A
A
A
B
A
0 , 10, 15 , 4, 25 , 6.6.a B b C c
Heron’s Formula
Let , , and be the sides of , and let denote the
( ) / 2. Then the area of is given by
Area - .
a b c ABC s
a b c ABC
s s a s b s c
semiperimeter
Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
Compute : (10 12 14) / 2 18.
Use Heron's Formula:
18 18 10 18 12 18 14
= 3456
=24 6 58.8
The area is approximately 58.8 square units.
s s
A
Chapter Test
3
2
1. Prove the identity cos3 4cos 3cos .
2. Write the expression in terms of sin and cos .
cos 2 sin 2
3. Find the general solution without using a calculator.
2cos 2 1
x x x
x x
x x
x
Chapter Test
4 2
2
4. Solve the equation graphically. Find all solutions in the interval [0,2 ).
sin 2
5. Find all solutions in the interval [0,2 ) without using a calculator.
sin 2sin 3 0
6. Solve the inequality.
x x
x x
Use any method, but give exact answers.
2cos 1 for 0 2
7. Solve , given 79 , 33 , and 7.
8. Find the area of , given 3, 5, and 6.
x x
ABC A B a
ABC a b c
Chapter Test
9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?
10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.
Chapter Test Solutions
2 2 3 2
3 3
3
2
cos3 cos(2 ) cos 2 cos sin 2 sin
cos sin cos 2sin cos sin cos 3cos sin
cos 3cos
1. Prove the identity cos3 4cos 3cos .
2. Write the expression in terms of sin and
1 cos 4cos 3cos .
x x x x x x x
x x x x x x x x x
x x
x x x
x x x
x
22 2
cos .
cos 2 sin 2
3. Find the general solution without using a calculator.
2co
1 4sin cos 2co
s 2 1
s sin
52 , 2
6 6
x
x x x x x
nx
x
n
Chapter Test Solutions
4 2
2
4. Solve the equation graphically. Find all solutions in the interval [0,2 ).
sin 2
5. Find all solutions in the interval [0,2 ) without using a calculator.
sin 2sin 3 0
6. Solve
1.15
3
2
x x
x
x
x
the inequality. Use any method, but give exact answers.
2cos 1 for 0 2
7. Solve
5,
3 3
6
, given 79 , 33 , and 7.
8. Find the area of ,
8 , 3.88, 6
given 3, 5,
.61
x x
ABC A B a
AB
C b
C a b
c
and 6. 7.5c
Chapter Test Solutions
9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?
10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.
≈0.6 mi
405π/24≈53.01