Download Geometry A Unit 9 Day 7 Notes Trigonometry and Triangle Area I

Geometry A Unit 9 Day 7 Notes
Trigonometry and Triangle Area
I. Explanation
A. We now have the ability to find sides and angles in non-right triangles. We do
not yet have a formula for the area of a triangle that does not require a right
triangle.
6
h
50o
15
II. SAS triangle area formula - _______________________________________
Examples:
Find the area of the triangles drawn below.
Ex. 1:
17
Ex. 2:
14
24
19
55o
11
Ex. 3:
36
71o
III. One other issue with the Law of Sines.
A. “The Ambiguity of Sine”
1. Using your calculator, find
sin 20o = _______________
2. Take all available decimal places of the answer to 1, and complete the
following:
sin-1 ________________________ = ______________
3. Using your calculator, find sin 160o = ____________________
4. Take all available decimal places of the answer to 3, and complete the
following:
sin-1 ______________________ = _____________
5. Using your calculator, find
cos 20o = _______________
6. Take all available decimal places of the answer to 1, and complete the
following:
cos-1 ______________________ = _______________
7. Using your calculator, find cos 160o = ____________________
8. Take all available decimal places of the answer to 3, and complete the
following:
cos-1 ______________________ = _______________
What we can take away from that is that we have to be careful when finding angles using the
Law of Sines because there are sometimes two possibilities for the missing angles.
B. Solve for all missing parts of the triangle by following the directions carefully.
C
C
5
6
A
5
6
B
10
1. Find  A using the Law of Cosines.
A
B
10
1. Find  C using the Law of Cosines.
2. Find  C using the Law of Sines.
2. Find  A using the Law of Sines.
3. Find  B using the Triangle Sum.
3. Find  B using the Triangle Sum.
Compare your answers in the left column with your answers in the right column from the
previous page.
Law of Cosines
Law of Cosines
A =
C =
Law of Sines
Law of Sines
C =
A =
Third Angle Theorem
Third Angle Theorem
B =
B =
What do you notice about  A?
What do you notice about  C?
What do you notice about  B?
What this means:
 Use the Law of Cosines whenever possible because it does NOT have an “Ambiguous
Case.”
 The only time you should need to worry about the Law of Sines “Ambiguous Case” is
when you are only given a SSA triangle (examples below).
 For a SSA case, it is possible that we can come up with two different angles, leading to
two completely different triangles.
Solve the triangle. You are given SSA – This is the ambiguous case.
a = 10
b = 16
30
HW: Finish CW Worksheet