6.1 Laws of Sines

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

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

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.

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.

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

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

The Laws of Sines

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

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

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

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

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

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

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

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

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

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

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

The Ambiguous Case Do you remember the Hinge Theorem from Geometry. Given two sides and one angle, two different triangles can be made. http://mrself.weebly.com/5-5-the-hinge-theorem.html

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

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

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

Find the missing Angles and Sides Given:

Find the missing Angles and Sides Given:

Find the missing Angles and Sides Given:

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

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