 Work out problems on board  Reminder about minutes/seconds at the end

The Law of Sines

We know that Trigonometry can help us solve right triangles. But not all triangles are right triangles. Fortunately, Trigonometry can help us solve non-right triangles as well. Non-right triangles are know as oblique triangles. There are two categories of oblique triangles—acute and obtuse.

4  SOHCAHTOA only works in RIGHT triangles!  How can we solve for unknowns in oblique triangles? B A C a c b

5  We wish to solve triangles which are not right triangles B A C a c b h Law of Sines!

The Law of Sines is used when we know any two angles and one side (ASA or AAS) or when we know two sides and an angle opposite one of those sides (ASS). Hints: *Create a proportion with only 1 unknown Ex: * Use 4 decimals

Ex 1: ASA. From the model, we need to determine a, b, and  (gamma) using the law of sines.

First off, 42º + 61º +  = 180º so that  = 77º. (Knowledge of two angles yields the third!)

Now by the law of sines, we have the following relationships:

Let’s solve for our unknowns:

Ex. 2: AAS From the model, we need to determine a, b, and  using the law of sines. Note:  + 110º + 40º = 180º so that  = 30º a b

By the law of sines, we have the following relationships:

Therefore,

The Ambiguous Case – ASS In this case, you may have information that results in one triangle, two triangles, or no triangles.

 If I know one of the possible angles, how do I find the other possible angle?  For example, if what solutions of x could be an angle in a triangle?  What do you notice about these two solutions? Supplementary! Between 0 and 180 deg.

 Or if what solutions of x could be an angle in a triangle?  What do you notice about these two solutions? Supplementary!

 The same is true for all other angle measures with the equivalent sine  So we can subtract the first angle from or 180 degrees to get the second angle.

Example #3: ASS Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

Thus, Therefore, there is no value for  that exists! No triangle is possible!

Example #4: ASS Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

So that, Interesting! Let’s see if one or both of these angle measures makes sense.

Case 1 Case 2 Both triangles are valid! Therefore, we have two possible cases to solve.

Finish Case 1:

Finish Case 2:

Wrapping it up, here are our two solutions:

Example #5: ASS: Two sides and an angle opposite one of the sides are given. Let’s try to solve this triangle.

By the law of sines,

Note: Only one is legitimate!

Thus, we have only one triangle. Now let’s find b.

By the law of sines,

Finally, we have:

The Area of a Triangle Using Trigonometry Given two sides and the included angle, can we find the area of the triangle? Remember

The Area of a Triangle Using Trigonometry We can find the area of a triangle if we are given any two sides of a triangle and the measure of the included angle. (SAS)

Example 7: Find the area of given a = 32 m, b = 9 m, and

x 15ft 15 º 65 º B A C SKIP???

 One min (single prime) is 1/60 of a degree  One second (double prime is 1/60 of a minute or 1/3600 of a degree.  Convert to degrees  40 + (20 * 1/60) + (50 * 1/60 * 1/60)

 6.1 Pg. 436 #1-7 odd, 13, 14, odd, 29, 31, 35, 36

 6.1 Pg. 436 #1-27 odd, 35