6 1 6 2 Law of Sines Area

6. 1 6. 2 Law of Sines & Area Formula

Objective: - To find missing parts of a triangle using the Law of Sines - To find the area of a triangle using side lengths

In a right triangle, we find missing angles and sides using : cos = adj sin = opp hyp Tan = opp adj

If you DO NOT have a right triangle, use the Law of sines or Law of Cosines to find missing parts.

Use the Law of Sines for CASE 1: ASA or AAS CASE 2: SSA

• The Sine Law is used when a side and the opposite angle are involved. A a

The sine law says : sin A = a sin B = sin C b c A c b C a B

The sine law says : sin A = a sin B = sin C b c You can also use: __a = b _ c__ = sin. A sin. B sin. C

Side a is opposite Side b is opposite Side c is opposite b C A c a B

ASA - The side is included between the 2 angles. b C A

AAS - The side is NOT included between the 2 angles. A C c

Find a don’t need sin A = sin Sin A B = ab B, b a A 0 101 C 0 48 a Sin C c 9

0 48 Sin A Sin C = a c 9 0 101 A 9 0 101 C 0 48 a

0 48 Sin A Sin C = a c 9 0 101 . 9816 =. 7431 A a 9 9 0 101 (9) (. 9816) =. 7431 a 0 8. 8 =. 7431 a 48 C a 11. 8 =a

0 48 Sin A Sin C = a c 9 0 101 (9) Sin 0 101 A = a Sin 0 101 sin 48 C 0 11. 9 48 = a a 0 48 9 sin 48

Round off error. In the 2 nd problem, there was no rounding until the last step.

In the case of SSA, there are sometimes 2 possible answers.

In the case of SSA, there are sometimes 2 possible answers.

Sometimes 1 ∆ can be formed

Sometimes a ∆ can’t be formed.

Find < A But we don’t know side a. If we find <C A 1 st, we will 12 10 have 2 <s. 0 52 C a Then get <A. B

Find < C 1 st A 12 10 0 52 B C a

139 41

Always calculate both. 0 =139 C is not possible. 0` 52 + 139 > 180 A 12 10 0 52 B C a

So C = 0 41 Now Find <A A = ( 180 - 52 - 41 0 A = 87 A 12 10 0 52 B C a 41 0 )

A 15 B 0 52 Find < A 12 C Find C 1 st

A 15 Find < A If we find <C 12 0 52 1 st, then we C have 2 <‘s. B

A 15 B 0 52 12 C 0 0 C = 80 or 100 Both are possible 0 0 A = 48 or 28

Lets investigate area of a triangle using Sine.

Area Formula A = 1/2 base * height = 1/2 b h B a c h A b C

If h is not known, it can be found : sin. C = h/a asin. C = h c A B h b a C

So if A = 1/2 b h substitute asin. C = h K = 1/2 a b sin. C just a different variable

When would you use this formula? if h is K= 1/2 ab sin. C unknow B c A b a C

Theorem: SAS Area Formula In any ∆ the area is 1/2 the product of any 2 sides and their included <.

Find Area K= 1/2 ab sin. C 0 K= 1/2 (6)(9) sin 29 B = 13. 1 6 c o 29 A C 9

Check by estimating. Remember Geometry: B c A 9 6 o 29 C

Longest side is opposite t biggest <. B c A 9 6 o 29 C

Shortest side is opposite t smallest <. B c A 9 6 o 29 C

Use estimation to see if your answers are reasonable.

Good luck.