Section 5-6 The Law of Sines

The Law of Sines can be used to solve triangles that are not right triangles. b c C a B a = b = c sin A sin B sin C

Solve ∆ABC if A=33°, B=105°, and b =37.9

Steps to solving triangles using Law of Sines Draw triangle Label parts Write the Law of Sines Substitute your values. Cross multiply Solve for the missing value.

Solve ∆LMN if L=29°, M=112°, and l =22 L N

Solve ∆LMN if L=29°, M=112°, and l =22 m sin 29 = 22 sin 112 m= 42.1 Still need values for N and n

N= 39° n= 28.6

Solve ∆ABC if A=41°, B=49°, and a =6.5 c b B a C

Solve ∆ABC if A=41°, B=49°, and a =6.5 c b 49° B 6.5 C C=90° b=7.5 c=9.9

Solve ∆ABC if A=80°, and a =12, b=19 c b B a C

Solve ∆ABC if A=41°, B=49°, and a =6.5 80° c 19 B 12 C C=90° b=7.5 c=9.9

Solve ∆ABC if A=106°, B=31°, and a =10 c b B a C

Solve ∆ABC if A=41°, B=49°, and a =6.5 106° c b 31° B 10 C C=90° b=7.5 c=9.9

John wants to measure the height of a tree John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33º. This particular tree grows at an angle of 83º with respect to the ground rather than vertically (90º). How tall is the tree?

A building is of unknown height A building is of unknown height. At a distance of 100 feet away from the building, an observer notices that the angle of elevation to the top of the building is 41º and that the angle of elevation to a poster on the side of the building is 21º. How far is the poster from the roof of the building?

An observer is near a river and wants to calculate the distance across the river. He measures the angle between his observations of two points on the shore, one on his side and one on the other side, to be 28º. The distance between him and the point on his side of the river can be measured and is 300 feet. The angle formed by him, the point on his side of the river, and the point directly on the opposite side of the river is 128º. What is the distance across the river?

Law of Sines the Ambiguous case When you have SSA you may have 1, 2 or no triangles.

Steps for applying the Law of Sines in Ambiguous Cases (SSA) Draw triangle Label parts Write the Law of Sines Substitute your values. Cross multiply & solve for the Sine of the missing angle. Solve for the missing angle measure by pressing 2nd Sine second answer.

The calculator will yield none or one If you get an “Error Domain” There is no triangle. If you get a number, write it down. This is the first solution for your angle measure. Take that first solution and subtract it from 180 to get the supplement. This is your second possible solution for your angle measure.

But you could have two solutions. Add the given angle and your first solution for the second angle. Subtract this sum from 1800. This is the measure of the 3rd angle. Add the given angle and your for the supplement of the second angle. If this sum is greater than 180 you do NOT have a second possible triangle. If this sum is less than 180 you DO have a second possible triangle. The 3rd angle of the second possible triangle is equal to 180 - (sum of the given angle and the supplement of the second angle).

Solving for the third side Now that you have 3 angles and 2 sides, repeat the same process: If you have two measurements for the second angle you will need to do this TWICE, once for each triangle.

More Law of Sines Word Problems A pig at Papas Barn just had a litter of piglets. The whole barn is about 12 miles long. In the middle of the night one piglet ran out of the barn a traveled down to the Bray Ranch which is about 13.5 miles away. When viewed on the towns map the two barns make a 115 degree angle. How far does the mama pig have to travel to get to her piglet.

A rocket tracking station has two telescopes A and B, placed 1 A rocket tracking station has two telescopes A and B, placed 1.4 miles apart. The telescopes lock onto a rocket and transmit their angles of elevation to a computer after a rocket launch. What is the distance to the rocket from telescope B at the moment when both tracking stations are directly east of the rocket telescope A reports an angle of elevation of 29 degrees and telescope B reports an angle of elevation of 49 degrees?

Two airplanes leave an airport at the same time Two airplanes leave an airport at the same time. One flies N 55 degrees W zt 340 mph, and the other flies S 30 degrees at 390 mph. How far apart are they after 2 hours?

A famous golfer tees off on a straight 380 yard par 4 and slices his drive to the right. The drive goes 280 yards from the tee. Using a 7-iron on his second shot, he hits the ball 160 yards and it lands inches from the hole. How many degrees (to the nearest degree) to the right of the line from the tee to the hole, did he slice his drive?