Transcript File

Algebra 2A
Chapter 13:
TRIGONOMETRY
θº
Lesson 13.1:
Right Triangle
Trigonometry
Learning Targets:
I can find values of the six trigonometric
functions (sine, cosine, tangent, cosecant,
secant, and cotangent) for acute angles.
I can solve equations involving inverse
trigonometric functions.
I can solve problems involving right
angles.
Six Trig Functions
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =
Example 1:
Find the value of the six
trigonometric functions
for angle G.
For this triangle, the leg opposite
to
and the leg adjacent
The hypotenuse is
Use opp = 24, adj
trigonometric ratio.
= 32, and hyp = 40 to write each
Example 1 (continued)
Answer:
Example 2:
Find the value of the six trigonometric
functions for angle A.
Answer:
Example 3:
Multiple-Choice Test Item
If
find the value of
A.
B.
C.
D.
csc A.
Example 3: (continued)
Read the Test Item
Draw a right triangle and label
one acute angle
and
leg
A. Since
, label the opposite
5 and the adjacent leg 3.
Example 3: (continued)
Solve the Test Item
Use the Pythagorean Theorem to find
c.
Pythagorean Theorem
Replace
a with 3 and b with 5.
Simplify.
Take the square root
of each side.
Example 3: (continued)
Now find
csc A.
Cosecant ratio
hyp with
and opp with 5.
Replace
Answer: D
Example 4:
Multiple-Choice Test Item
If
find the value of
A
B
C
D
Answer: C
cos B.
I can solve problems involving
right angles.
Example 5:
“Solve” means to find ALL
of the side lengths and
angle measures.
Solve XYZ. Round measures of
sides to the nearest tenth and
measures of angles to the nearest
degree.
x
z
You know the measures of one side, one acute angle, and the right
angle. You need to find x, z, and Y.
Example 6:
Solve XYZ. Round measures of sides to
the nearest tenth and measures of angles
to the nearest degree.
Answer:
Example 7:
Solve ABC. Round measures of
sides to the nearest tenth and
measures of angles to the nearest
degree.
You know the measures of the sides. You need to
find A and B.
Example 8:
Solve ABC. Round measures of
sides to the nearest tenth and
measures of angles to the nearest
degree.
Answer:
Example 9:
Bridge Construction In order to
construct a bridge across a river, the
width of the river must be determined.
A stake is planted on one side of the
river directly across from a second
stake on the opposite side. At a
distance 30 meters to the left of the
stake, an angle of 55 is measured
between the two stakes. Find the width
of the river.
Angles of
Elevation/Depression
Angle of Depression
θº
θº
Angle of Elevation
Example 10:
Skiing A run has an angle of elevation of 15.7 and a
vertical drop of 1800 feet. Estimate the length of this
run.
Let
represent the length of the run. Write an
equation using a trigonometric function that
involves the ratio of
and 1800.
Independent Practice:


Read 13.1 pages 701-705
Complete Worksheet 13.1
Warm - Up
about 2.4°
Lesson 13.4:
The Law of Sines
Learning Target:

I can solve problems by using
the Law of Sines.
The Law of Sines
In ANY triangle
ABC ,
C
b
A
sin A sin B sin C


a
b
c
a
c
B
When to Use the Law of Sines:
The Law of Sines is
especially useful when
solving triangles given
AAS
ASA
SSA
____, ____, or ____.
Example 1:
Find b.
Example 2:
Find D.
Application
Example 3:
When the sun’s angle of elevation is 76°, a tree tilted at
an angle of 4° from the vertical casts an 18-foot shadow.
Find the height of the tree, to the nearest tenth of a foot.
Application
Example 4:
A ranger tower at point A is 42 kilometers north of a
ranger tower at point B. A fire at point C is observed
from both towers. If BAC measures 43° and ABC
measures 68°, which ranger tower is closer to the fire?
Independent Practice:


Read 13.4 pages 725-729
Complete Worksheet 13.4
Lesson 13.5:
The Law of Cosines
Learning Targets:

I can solve problems by using
the Law of Cosines.
I can determine whether a triangle
can be solved by first using the Law
of Sines or the Law of Cosines.
The Law of Cosines
In ANY triangle
ABC ,
2 = a2 + b2– 2ab∙cos C
c
_____________________________
a2
b2
c2–
=
+
2bc∙cos
A
_____________________________
2 = a2 + c2– 2ac∙cos B
b
_____________________________
C
a
b
A
c
B
When to Use the Law of Cosines:
The Law of Cosines is
especially useful when
solving triangles given
SAS
SSS
____ or ____.
Using the Law of Cosines
to Find the Missing Side
SAS
Example 1:
Find c.
Example 2:
S
Find r.
12
R
62°
14
T
Application
Example 4
A ranger tower at point A is directly north of a ranger
tower at point B. A fire at point C is observed from both
towers. The distance from the fire to tower A is 60 miles,
and the distance from the fire to tower B is 50 miles.
If mACB  62 , find the distance between the towers.
Using the Law of Cosines
to Find the Missing Angle
SSS
Example 4:
E
Find D.
7
5
F
D
8
Example 5:
Find L.
K
15
15
L
19
J
Independent Practice:


Read 13.5 pages 733-735
Complete Worksheet 13.5