1. 6.1 Laws of Sines

2. The Laws of Sine can be used with Oblique triangle
Oblique triangle is a triangle that contains no right angle.

3. What we already know The interior angles total 180.
We can’t use the Pythagorean Theorem. Why not?
For later, area = ½ bh
Larger angles are across from longer sides and vice versa.
The sum of two smaller sides must be greater than the third. A B C a b c

4. There are three possible configurations that will enable us to use the Law of Sines. They are shown below.
You don’t have an angle and side opposite it here but can easily find the angle opposite the side you know since the sum of the angles in a triangle must be 180°.
ASA
SAA
You may have an angle, a side and then another angle
You may have a side and then an angle and then another angle
What this means is that you need to already know an angle and a side opposite it (and one other side or angle) to use the Law of Sines.
SSA
You may have two sides and then an angle not between them.

5. General Process for Law Of Sines
Except for the ASA triangle, you will always have enough information for 1 full fraction and half of another. Start with that to find a fourth piece of data.
Once you know 2 angles, you can subtract from 180 to find the 3rd.
To avoid rounding error, use given data instead of computed data whenever possible.

6. Use Law of SINES when ... AAS - 2 angles and 1 adjacent side
…you have 3 parts of a triangle and you need to find the other 3 parts.
They cannot be just ANY 3 dimensions though, or you won’t have enough info to solve the Law of Sines equation.
Use the Law of Sines if you are given:
AAS - 2 angles and 1 adjacent side
ASA - 2 angles and their included side
ASS – (SOMETIMES) 2 sides and their adjacent angle

7. Draw a perpendicular line and call the length h
Draw a perpendicular line and call the length h. We do this so that we have a right triangle which we already know how to work with.  c a h   b
Let’s write some trig functions we know from the right triangles formed.
Solve these for h
Since these both = h we can substitute
divide both sides by ac ac ac

8. The Laws of Sines

9. Using the Law of Sines
Given: How do you find angle B?

10. Using the Law of Sines
Given: How do you find side b?

11. Using the Law of Sines
Given: How do you find side b?

12. Using the Law of Sines
Given: How do you find side b?

13. Using the Law of Sines
Given: How do you find side c?

14. Using the Law of Sines
Given: How do you find side c?

15. The Ambiguous Case Look at this triangle. If we look at where angle A
Is Acute

16. The Ambiguous Case Look at this triangle. If we look at
If a = h, then there is one triangle

17. The Ambiguous Case Look at this triangle. If we look at
If a < h, then there is no triangle

18. The Ambiguous Case Look at this triangle. If we look at
If a > b, then there is one triangle

19. The Ambiguous Case Look at this triangle. If we look at
If h< a <b, then there is two triangles

20. The Ambiguous Case Do you remember the Hinge Theorem from Geometry.
Given two sides and one angle, two different triangles can be made.

21. The Ambiguous Case Where Angle A is Obtuse. If a ≤ b, there is no
triangle

22. The Ambiguous Case Where Angle A is Obtuse. If a > b, there is one
triangle

23. Area of an Oblique triangle
Using two sides and an Angle.

24. Find the missing Angles and Sides
Given:

25. Find the missing Angles and Sides
Given:

26. Find the missing Angles and Sides
Given:

27. Homework
Page 416
# 1, 7, 13, 19,
25, 31, 37,
43, 49

28. Homework
Page 416
# 4, 10, 16, 22,
28, 34, 40,
46, 52