Download using the Law of Sines. - Fort Thomas Independent Schools

P.o.D. – Find the amplitude,
period, and vertical translation
of each graph.
1.) 𝑦 = 3 sin(2𝑥 ) − 5
2.) 𝑦 = cos(𝑥)
3.) 𝑦 = −4 sin(. 5𝑥 ) + 2
4.) 𝑦 = −2 cos(3𝑥 ) − 6
1
1
3
3
5.) 𝑦 = sin ( 𝑥) +
6.) 𝑦 = − cos(6𝑥 ) − 7
7.) 𝑦 = sin(𝑥 ) + 𝜋
10-7: The Law of Sines
Learning Target(s): I can
determine the measure of an
angle given its sine, cosine, or
tangent; find the missing sides
and angles of a triangle using the
Law of Sines; identify and use
theorems relating sines and
cosines; solve real-world
problems using the Law of Sines.
Essential Question: How do you
use trigonometry to solve and
find the areas of oblique
triangles?
Oblique Triangle = a triangle
that has no right angle; could be
acute or obtuse.
Law of Sines:
C
a
A
c
B
EX: For triangle ABC, A=30
degrees, B=45 degrees, and a=32
feet. Find the remaining sides
and angle.
Always begin by drawing a
picture.
B
45
32
30
A
C
The easiest thing to find is the
missing angle.
Now, use the given information
to find side b.
Next, find side c.
Finally, write your solution
with all 6 parts of the triangle
together.
EX: Because of prevailing winds,
a tree grew so that it was
leaning 6 degrees from the
vertical. At a point 30 meters
from the tree, the angle of
elevation to the top of the tree is
22.5 degrees. Find the height of
the tree.
h
96
22.5
30
Begin by finding the missing
angle.
Set up the Law of Sine to find h.
EX: For the triangle ABC,
a=12inches, b=5inches, and A=31
degrees. Find the remaining
sides and angles.
We need to find angle B.
Next, find angle C by
subtraction.
Now, find side c using the Law
of Sines.
Write your final answer.
The Ambiguous Case of the Law
of Sines (SSA):
Why couldn’t you prove triangle
congruency with SSA in
geometry?
(Draw a picture on the
whiteboard)
Three possibilities will exist:
Case I: If A is an acute angle:
If 𝑎 < 𝑏 sin 𝐴, then ___ triangle
exists.
If 𝑎 = 𝑏 sin 𝐴, then exactly ____
right triangle exists.
If 𝑎 > 𝑏 sin 𝐴, then ___ triangles
exist.
If 𝑎 ≥ 𝑏, then ____ triangle exists.
Case II: If A is an obtuse angle:
If a < b, then ___ triangle exists.
If a > b, then ____ triangle exists.
EX: Show that there is no
triangle for which A=60degrees,
a=4, and b=14.
We are in Case I since A is acute.
In Case I, if a<b sin A, then
EX: Find two triangles for
which A=58degrees, a=4.5, and
b=5.
Again, we are in Case I since A is
acute. We now need to find b sin
A.
Since a>b sin A {4.5>4.24}, we
know that we have ____
solutions. Begin by solving using
the Law of Sines.
Triangle #1:
Use subtraction to find angle C.
Now, use the Law of Sines to
find side c.
Therefore, the first triangle
consists of:
Triangle #2:
In other words, the second angle
B is the ______________ of the first
angle B.
Find angle 𝐶2 by subtraction.
Lastly, find side 𝑐2 using the Law
of Sines.
Triangle #2 consists of:
Area of a Triangle:
The area of any triangle is onehalf the product of the lengths
of two sides times the sine of
their included angle. That is,
EX: Find the area of a triangular
lot containing the side lengths
that measure 24 yards and 18
yards and form an angle of 80
degrees.
*On your own, write a
calculator program titled
SASAREA that will compute the
area of a triangle given two sides
and the included angle.
EX: On a small lake, a child
swam from point A to point B at
a bearing of 𝑁 28° 𝐸. The child
then swam to point C at a
bearing of 𝑁 58° 𝑊. Point C is 800
meters due north of point A.
How many total meters did the
child swim?
(draw a picture and show the
work on the whiteboard)
EX: Suppose you are given a
triangle with two angles of 42
degrees and 74 degrees and an
included side of 18-m. Find the
length of the side opposite the
74 degrees.
(draw a triangle on the
whiteboard and have a student
come to the board to solve)
EX: In triangle TRI, angle R is 25
degrees, angle I is 72 degrees,
and side r is 13. Find side t.
(draw a triangle on the board
and have a student come to the
board to solve)
HW Pg.703 2, 7-12, 15-18
Worksheet 10.7B