Recommended Homework: 4.2: 4-7, 17-23 Odd, 25-41, 43, 46-50, 53

Law of Sines P. 1

4.2 Law Of Sines

Recommended Homework: 4.2: 4-7, 17-23 odd, 25-41, 43, 46-50, 53

Oblique triangles have NO Right angles. In order to solve an oblique triangle, we must know the measure of 1 side and the measure of 2 other parts. If we know 2 angles and one side or 2 sides and the angle opposite one of the sides then we can use the law of sines.

sin α = h/b  h = b sinα sin β = h/a  h = a sinβ

sin sin So bsin = a sin or  α β a b In General, the sin of an angle over its opposite side = sin of an another angle over its opposite side

This is the Law of Sines sin  sin  sin a b c Law of Sines P. 2

This formula is used for SSA, ASA, and AAS problems.

Remember also, the sum of the angles in a triangle =

Case 1: Angle, Angle Side (AAS) or Angle, Side Angle (ASA) Strategy: 1. Find the missing angle using 180º-(sum of other angles) 2. Find the missing side using the Law of Sines 3. Find the remaining side using the Law of Sines

Example: b = 10, β = 106.5, γ = 23.5 Law of Sines P. 3

Example: α = 15, β = 105, c = 23 Law of Sines P. 4

Case 2: Side, Side Angle (SSA) {ambiguous case} There are really 3 possible cases – either there is a unique triangle, no triangle or 2 triangles that satisfy the given information. Because sin(angle) is positive in QI AND QII, it may be possible to get more than one answer. ALWAYS TRY FOR 2 solutions . Strategy: 1. Find the missing angle using the law of sines (NOTE: The range of the sine function is [-1, 1]. If the sine of the angle is <-1 or >1, there is no triangle.) 2. Find the remaining angle using 180º - (sum of the other angles) 3. Find the missing side using the Law of Sines. 4. ***VERY IMPORTANT!!!*** Check to see if there are 2 triangles a. Find the QII angle from #1 (180º- ) b. Add the new angle in step a and the given angle i. If the sum is < 180º, there are 2 triangles ii. If the sum is > 180º, there is only one triangle c. If there are 2 triangles, solve the remaining triangle using the QII angle (steps 2 & 3). Law of Sines P. 5

Example: b = 10, a = 25, and α = 42 Law of Sines P. 6

Example: a = 22, b = 31, α = 85

Example: a = 12, b = 31, α = 21.1 Law of Sines P. 7

Applications: Example: The angle of elevation from 2 towns A and B to a plane up in the air are 23 and 15 respectively. The towns are 4 miles apart and both are west of the plane. a) Find the distance from town A to the plane b) Find the altitude of the plane. Law of Sines P. 8

Example: A hiker is standing x feet away from the tree’s base of a tree looks up and sees the top of the tree at an angle of elevation of 44. He goes another 100 feet away and takes another sighting. This time the angle of elevation to the top of the tree is 37.2. What is the height of the tree? Law of Sines P. 9

Review of Bearings: N

Example: A plane flies 500km with a bearing of N44ºW from Naples to Elgin. The plane then flies 720km from Elgin to Canton. Find the bearing of the flight from Elgin to Canton if Canton is due West of Naples.

Canton Naples Law of Sines P. 10

Example: A family is traveling due west on a road that passes a famous landmark. At a given time, the bearing to the landmark is N62ºW, and after traveling 5 miles farther, the bearing is N38ºW. What is the closest the family will come to the landmark while on the road? Law of Sines P. 11

Example: A bridge is to be built across a small lake from a condo to a club house on the other side of the lake. The bearing from the condo to the club house is S40ºW. From a tree 100 m from the condo, the bearings to the condo and club house are S72ºE and S26ºE respectively. Find the distance from the condo to the clubhouse. Law of Sines P. 12

Example: The bearing from Bear fire tower to Smokey fire tower is N70ºE. The two fire towers are 30 km apart. A fire is spotted by rangers in each tower. The bearing to the fire from Bear tower is N85ºE and from Smokey tower is S75ºE. Find the distance of the fire to each tower. Law of Sines P. 13

Example: Coast Guard Station Peace is located 120 mi due west of Station Safety. A ship at sea sends an SOS call that is received by each station. The call to station Peace indicates the bearing of the ship from Peace is N40ºE. The call to station Safety indicates that the bearing of the ship from Safety is N30ºW. How far is the ship from each station?