Assignment
• Trig Ratios III
Worksheets (Online)
• Challenge Problem:
Find a formula for the
area of a triangle given
a, b, and  .
SohCahToa
Soh
sin  opposite
hypotenuse
Cah
cos  adjacent
hypotenuse
Toa
tan  opposite
adjacent
Example 1
Find the area of ΔABC.
Trigonometric Ratios III
Objectives:
1. To derive the Laws of Sines and Cosines
2. To find all the angles and sides of any
triangle using the Laws of Sines and
Cosines
Investigation: Law of Sines
As the previous example illustrates,
trigonometry can be applied to non-right, or
oblique, triangles. In that example, we
used it to find an unknown height. Now,
we’ll use it to find a missing side length of
a non-right triangle.
Investigation: Law of Sines
Step 1: Draw ΔABC
as shown with
height h. Notice
that side a is
opposite A, b is
opposite B, and c
is opposite C.
Investigation: Law of Sines
Step 2: Write an
equation for h using
sin(A).
Step 3: Write an
equation for h using
sin(B).
Investigation: Law of Sines
Step 4: Use
substitution to
combine the two
previous equations
and write your final
equation as a
proportion.
Investigation: Law of Sines
Step 5: Write an
equation for k using
sin(B).
Step 6: Write an
equation for k using
sin(C).
Investigation: Law of Sines
Step 7: Use
substitution to
combine the two
previous equations
and write your final
equation as a
proportion.
Investigation: Law of Sines
Step 8: Finally, use
the transitive
property to combine
the proportion from
S4 and S7. This is
the Law of Sines.
Law of Sines
If ΔABC has side lengths a, b, and c as
shown, then
Looking for an ANGLE:
sin A sin B sin C


a
b
c
Looking for a SIDE:
a
b
c


sin A sin B sin C
Example 2
Find the length of side
AC in ΔABC.
Example 3
Find the length of side
AC in ΔABC.
Acute or Obtuse?
The Law of Sines can also be used to find a
missing angle measure, but only if you
know that it is acute or obtuse. This is
simply because SSA is not a congruence
shortcut.
C
160
C
260
260
160
36
B1
36
A
B2
A
Example 4
Solve ΔABC.
Example 5
Find the indicated measure.
1. x and y
2.  and 
Example 6
Find the length of AC in acute triangle ABC.
Law of Sines
As the previous example
demonstrates, you cannot
always use the Law of
Sines for every triangle.
You need either two sides
and an angle or two
angles and a side in the
following configurations:
ASA
AAS
SSA
Law of Cosines
To derive the Law of
Cosines, we need an
interesting diagram
like the one shown.
Click the picture for a
Geometer’s
Sketchpad
Demonstration of this
useful equation.
Law of Cosines
If ABC has sides of
length a, b, and c as
shown, then
c 2  a 2  b 2  2ab  cos C
b 2  a 2  c 2  2ac  cos B
a 2  b 2  c 2  2bc  cos A
Example 7
Simplify the equation below for C = 90°.
c 2  a 2  b 2  2ab  cos C
Example 8
Solve the equation below for C.
c 2  a 2  b 2  2ab  cos C
Example 9
Find the length of AC in acute triangle ABC.
Example 10
Solve ΔABC.
Pro Tip: SSS
When using the Law of
Cosines to find a missing
angle (SSS), it’s a good
idea to find the angle
opposite the longest side
first. This is just in case
the angle turns out to be
obtuse. Regardless of
what type of angle this
turns out to be, use the
Law of Sines and the
Triangle Sum Theorem to
find the other two angles.
1: Use the Law of Cosines
2: Use the Law of Sines
3: Use the Triangle Sum
Example 11
Find the indicated measure.
1. x and y
2.  and 
Summary
Law of
Sines
Law of
Cosines
• ASA
• AAS
• SSA
• SAS
• SSS
Given three pieces of
any triangle, you
can use the Law of
Sines or the Law of
Cosines to
completely solve
the triangle.
Assignment
• Trig Ratios III
Worksheets (Online)
• Challenge Problem:
Find a formula for the
area of a triangle given
a, b, and  .