Work out problems on board Add visuals (ranges of arccos and arcsin) to show why you use LOC for big angles and LOS for small angles

6.2 The Law of Cosines

Which proved triangles congruent in Geometry? SSS ASA AAS SAS AAA ASS

The same ones that define a specific triangle! SSS - congruent ASA - congruent AAS – congruent SAS – congruent AAA – not congruent ASS – not congruent

Which proved triangles congruent in Geometry? SSS - congruent ASA – congruent – Solve w/ Law of Sines AAS – congruent – Solve w/ Law of Sines SAS – congruent AAA – not congruent ASS – not congruent

6 Solving an SAS Triangle The Law of Sines was good for –ASA- two angles and the included side –AAS- two angles and any side –SSA- two sides and an opposite angle (being aware of possible ambiguity) Why would the Law of Sines not work for an SAS triangle? ° No side opposite from any angle to get the ratio

Let's consider types of triangles with the three pieces of information shown below. SAS You may have a side, an angle, and then another side AAA You may have all three angles. SSS You may have all three sides This case doesn't determine a triangle because similar triangles have the same angles and shape but "blown up" or "shrunk down" We can't use the Law of Sines on these because we don't have an angle and a side opposite it. We need another method for SAS and SSS triangles. AAA

LAW OF COSINES Do you see a pattern?

9 Deriving the Law of Cosines Write an equation using Pythagorean theorem for shaded triangle that only includes sides and angles of the oblique triangle. b h a k c - k A B C c

Since the Law of Cosines is more involved than the Law of Sines, when you see a triangle to solve you first look to see if you have an angle (or can find one) and a side opposite it. You can do this for ASA, AAS and SSA. In these cases you'd solve using the Law of Sines. However, if the 3 pieces of info you know don't include an angle and side opposite it, you must use the Law of Cosines. These would be for SAS and SSS (remember you can't solve for AAA).

Ex. 1: Solve a triangle where b = 1, c = 3 and A = 80° Draw a picture. 80 B C a 1 3 Do we know an angle and side opposite it? No so we must use Law of Cosines. Hint: we will be solving for the side opposite the angle we know. This is SAS times the cosine of the angle between those sides One side squared sum of each of the other sides squared minus 2 times the product of those other sides Now punch buttons on your calculator to find a. It will be square root of right hand side. a = CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction

We'll label side a with the value we found. We now have all of the sides but how can we find an angle? 80 B C Hint: We have an angle and a side opposite it C is easy to find since the sum of the angles is a triangle is 180° When taking arcsin, use 2 nd answer on your calculator for accuracy!

Ex. 2: Solve a triangle where a = 5, b = 8 and c = 9 Draw a picture. B C Do we know an angle and side opposite it? No, so we must use Law of Cosines. Let's choose to find angle C first. This is SSS times the cosine of the angle between those sides One side squared sum of each of the other sides squared minus 2 times the product of those other sides CAUTION: Don't forget order of operations: powers then multiplication BEFORE addition and subtraction A 84.26

How can we find one of the remaining angles? B  Do we know an angle and side opposite it? A Yes, so use Law of Sines.

Too easy, what’s the catch? After we use L.O.C. we need to use law of sines to find the remaining sides and angles. The range of arcsin is -90 deg to 90 deg, but what if the angle is obtuse? Then taking the arcsin won’t get us the correct angle! To avoid this problem – When using L.O.S. after L.O.C. always find the smallest angle FIRST The smallest angle has to be acute since there can’t be more than one obtuse angle in a triangle. Then use the triangle sum thm to find the 3 rd angle.

16 Try it on your own! #1 Find the three angles of the triangle ABC if 8 6 A B C 12

17 Try it on your own! #2 Find the remaining angles and side of the triangle ABC if A B C 12

18 Summary – What could we use to solve the following triangles? Uh, nothing. It’s AAA

ASA – although we don’t know an angle and side opposite each other we can find the 3 rd angle then do law of sines Summary – What could we use to solve the following triangles? Do we know an angle and side opposite it? Could we find it?

AAS – law of sines Summary – What could we use to solve the following triangles? Do we know an angle and side opposite it?

ASS, we can use law of sines but need to check for 1, 2 or no triangles. Summary – What could we use to solve the following triangles? Do we know an angle and side opposite it?

SAS – don’t know (and can’t find) angle and side opposite Law of Cosines Summary – What could we use to solve the following triangles? Do we know an angle and side opposite it?

SSS – don’t know (and can’t find) angle and side opposite Law of Cosines Summary – What could we use to solve the following triangles? Do we know an angle and side opposite it?

24 Wing Span The leading edge of each wing of the B-2 Stealth Bomber measures feet in length. The angle between the wing's leading edges is °. What is the wing span (the distance from A to C)? Note these are the actual dimensions! A C

25 Wing Span A C

Navigational Bearings The direction to a point is stated as the number of degrees east or west of north or south. For example, the direction of A from O is N30ºE. B is N60ºW from O. C is S70ºE from O. D is S80ºW from O

H Dub 6-2 Pg. 443 #2-16even, 17-22all, 29, 34, 35

Practice #1 Practice #2

LAW OF COSINES Use these to find missing sides Use these to find missing angles Do you see a pattern?

Practice #1

Practice #2