THE LAW OF SINES states the following:The sides of a triangle are proportional to one another in the

same ratio as the sines of their opposite angles

This means that in the oblique triangle ABC, side a, for example, is to side b as the sine of angle A is to the sine of angle B.

a  =  sin A similarly b = sin Bb sin B c sin C

You sometimes see the law of sines stated algebraically as

sin A = sin B = sin C or a = b = c .

a b c sin A sin B sin C

THE LAW OF SINES

a)  The three angles of a triangle are 40°, 75°, and 65°.  In what ratio are the three sides? Sketch the figure and place the ratio numbers.

Solution.  To find the ratios of the sides, we must evaluate the sines of their opposite angles.

sin 40° = .643sin 75° = .966sin 65° = .906These are the ratios of the sides opposite those

angles:

Example 1.

B)  When the side opposite the 75° angle is 10 cm, how long is the side opposite the 40° angle?

Solution.  Let us call that side x.  Now, according to the Law of Sines, in every triangle with those angles, the sides are in the ratio 643 : 966 : 906. Therefore,

  x  = 643 10 966

x =6.656 cm

The three angles of a triangle are 105°, 25°, and 50°.  In what ratio are a)  the sides?  Sketch the triangle.

b)  If the side opposite 25° is 10 cm, how long is the side opposite 50°?

Checkpoint

When to use the law of sines formula?when you know 2 sides and an angle (case 1) and

you want to find the measure of an angle opposite a known side.

when you know 2 angles and 1 side and want to get the side opposite a known angle (case 2).

In both cases, you must already know a side and an angle that are opposite of each other.

Cases when you can not use the Law of Sines

The picture below illustrates a case not suited for the law of sines. Since we do not know an opposite side and angle, we cannot employ the law of sines formula.

Example 2. Use the Law of Sines formula to solve for b

Is it possible to use the law of sines to calculate x pictured in the triangle below?

Answer:Yes, first you must remember that the

sum of the interior angles of a triangle is 180 in order to calculate the measure of the angle opposite of the side of length 19. Now that we have the measure of that angle, use the law of sines to find value of x

Critical Thinking

Answer:

When there are no Triangles!

So if you are given any 2 sides and an angle, can you always use the law of sines to find all the other sides and angles of a triangle?

The answer is Not always!

Consider the following. Let's imagine that we know that there is some 'triangle' ABC with the following information: BC = 23 ,AC = 3, angle B = 44.

Therefore no triangle can be drawn with these

givens.

SAS - Side, Angle, SideASA - Angle, Side, AngleAAS - Angle, Angle, SideSSS - Side, Side, SideHL – Hypotenuse Leg for Right Triangles

We also discovered that SSA did not work to prove triangles congruent.We politely called it the Donkey Theorem  ; - )

In Geometry, we found that we could prove two triangles congruent using:

By definition, the word ambiguous means open to two or more interpretations.Such is the case for certain solutions when working with the Law of Sines.

  •  ASA or AAS,   the Law of Sines will nicely provide you with ONE solution for a missing side.

SSA (where you must find an angle), the Law of Sines could possibly provide you with one , 2 solutions or even no solution.

Case I. How many distinct triangles can be formed from the triangle

Case II. How many distinct triangles can be formed from the triangle ?

Case III. How many distinct triangles can be formed from the triangle ?

The law of cosines for calculating one side of a triangle when the angle opposite and the other two sides are known. It can be used in conjunction with the law of sines to find all sides and angles. a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos Bc2 = a2 + b2 – 2ab cos C Cos A = b2 + c2 - a2

2bcCos B = a2 + c2 - b2

2acCos C = a2 + b2 - c2

2ab

The law of cosines

Find the angles of triangle RST

s = 70 ; t = 40.5 ; r = 90

Example 2.

A ship R leaves port P and travels at the rate of 16 mph. Another ship Q leaves the same port at the same time and travels at 12mph. Both ships are traveling in straight paths. The directions taken by the two ships make an angle of 55 degrees. How far apart are the ships after 3 hours?

Example 3.

You are heading to Beech Mountain for a ski trip. Unfortunately, state road 105 in North Carolina is blocked off due to a chemical spill. You have to get to Tynecastle Highway which leads to the resort at which you are staying. NC-105 would get you to Tynecastle Hwy in 12.8 miles. The detour begins with a 18o  veer off onto a road that runs through the local city. After 6 miles, there is another turn that leads to Tynecastle Hwy. Assuming that both roads on the detour are straight, how many extra miles are you traveling to reach your destination?

Find the remaining parts of the triangle1. b= 12 ; c = 27; a = 24

2. To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake?

Launch- Nov. 3

296 yards

After the hurricane, the small tree in my neighbor’s yard was leaning. To keep it from falling, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3½ feet above the ground. Determine the angle formed from the tree to the ground.

106.1º